Inclusion Criteria Step 1:

          -  Male or female patients 18 years or older.

          -  Patients must have locally advanced, metastatic or refractory leiomyosarcoma or
             dedifferentiated liposarcoma.

          -  A) Patients enrolled into the dedifferentiated cohort do not require prior systemic
             therapy (may be naive to systemic therapy). B) Leiomyosarcoma patients must have had
             at least 1 prior systemic therapy (does not include adjuvant/neoadjuvant therapy in a
             curative setting). There is no limits on prior number of therapies for either cohort.

          -  Measurable disease per RECIST 1.1

          -  Radiological or objective evidence of recurrence or progression on or after the last
             systemic therapy prior to enrollment.

          -  Time since the last prior therapy to treat underlying malignancy to start of drug:

               -  Cytotoxic chemotherapy: greater than the duration of the most recent cycle of the
                  previous regimen (with a minimum of two weeks for all)

               -  Biologic therapy (e.g., antibodies): ≥ four weeks

               -  ≥ 5 x t1/2 of a small molecule therapeutic, not otherwise defined above, with a
                  minimum of 2 weeks (including aromatase inhibitors and tamoxifen).

          -  Eastern Cooperative Oncology Group (ECOG) performance status 0-1.

          -  Patient has signed the Informed Consent (ICF) prior to any screening procedures being
             performed and is able to comply with protocol requirements.

          -  Patient has adequate bone marrow and organ function as defined by the following
             laboratory values at screening:

               -  Absolute neutrophil count ≥1.5 × 109/L

               -  Platelets ≥100 × 109/L

               -  Hemoglobin ≥9.0 g/dL

               -  Potassium, total calcium (corrected for serum albumin), magnesium, sodium and
                  phosphorus within normal limits for the institution or corrected to within normal
                  limits with supplements before first dose of study medication

               -  INR ≤1.5

               -  Serum creatinine <1.5 mg/dL or creatinine clearance ≥50 mL/min

               -  In the absence of liver metastases, alanine aminotransferase (ALT) and aspartate
                  aminotransferase (AST) <2.5 x ULN. If the patient has liver metastases, ALT and
                  AST <5 x ULN

               -  Total bilirubin < ULN; or total bilirubin ≤3.0 x ULN or direct bilirubin ≤1.5 x
                  ULN in patients with well-documented Gilbert's Syndrome.

               -  Fasting plasma glucose <140 mg/dL / 7.7 mmol/L and Glycosylated Hemoglobin
                  (HbA1c) ≤ 8% (both criteria have to be met)

          -  Fasting serum cholesterol ≤ 300 mg/dl or 7.75 mmol/L and fasting triglycerides ≤ 2.5 ×
             ULN. In case one or both of these thresholds are exceeded, the patient can only be
             included after initiation of statin therapy and when the above mentioned values have
             been achieved.

          -  Must be able to swallow ribociclib and everolimus capsules/tablets.

        Exclusion Criteria Step 1:

          -  Patient has a known hypersensitivity to any of the excipients of Ribociclib or
             Everolimus.

          -  Previous treatment with CDK4/6 inhibitors or mTOR inhibitors.

          -  Patient has a concurrent malignancy or malignancy within 3 years prior to starting
             study drug, with the exception of adequately treated, basal or squamous cell
             carcinoma, non-melanomatous skin cancer or curatively resected cervical cancer.

          -  Patients with central nervous system (CNS) involvement unless they meet ALL of the
             following criteria:

               -  At least 4 weeks from prior therapy completion (including radiation and/or
                  surgery) to starting the study treatment

               -  Clinically stable CNS tumor at the time of screening and not receiving steroids
                  and/or enzyme- inducing anti-epileptic medications for brain metastases.

          -  Patient has impairment of gastrointestinal (GI) function or GI disease that may
             significantly alter the absorption of the study drugs (e.g., ulcerative diseases,
             uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome, or small bowel
             resection).

          -  Patient has a known history of HIV infection (testing not mandatory).

          -  Patient has any other concurrent severe and/or uncontrolled medical condition that
             would, in the investigator's judgment, cause unacceptable safety risks, contraindicate
             patient participation in the clinical study or compromise compliance with the protocol
             (e.g. chronic pancreatitis, chronic active hepatitis, active untreated or uncontrolled
             fungal, bacterial or viral infections, etc.).

          -  Clinically significant, uncontrolled heart disease and/or cardiac repolarization
             abnormality, including any of the following:

               -  History of angina pectoris, symptomatic pericarditis, coronary artery bypass
                  graft (CABG) or myocardial infarction within 6 months prior to study entry.

               -  Documented cardiomyopathy

               -  Left Ventricular Ejection Fraction (LVEF) <50% as determined by Multiple Gated
                  acquisition (MUGA) scan or echocardiogram (ECHO) at screening

               -  History of cardiac failure, significant/symptomatic bradycardia, Long QT
                  syndrome, family history of idiopathic sudden death or congenital long QT
                  syndrome or any of the following:

                    -  Known risk to prolong the QT interval or induce Torsade's de Pointes.

                    -  Uncorrected hypomagnesemia or hypokalemia.

                    -  Systolic Blood Pressure (SBP) >160 mmHg or <90 mmHg.

                    -  Bradycardia (heart rate <50 at rest), by ECG or pulse.

                    -  On screening, inability to determine the QTcF interval on the ECG (i.e.:
                       unreadable or not interpretable) or QTcF >450 (based on a mean of 3 ECGs).

          -  Patient is currently receiving any of the following medications and cannot be
             discontinued 7 days prior to starting study drug (see Table 4 for details):

               -  Known strong inducers or inhibitors of CYP3A4/5, including grapefruit, grapefruit
                  hybrids, pummelos, star-fruit, and Seville oranges

               -  That have a narrow therapeutic window and are predominantly metabolized through
                  CYP3A4/5

               -  Herbal preparations/medications, dietary supplements. Acceptable supplements
                  include multivitamins, vitamin D and calcium

               -  Angiotensin-converting enzyme (ACE) inhibitor therapy

          -  Receipt of a live vaccine within 30 days prior to starting study drug. Examples of
             live vaccines are: intranasal influenza, measles, mumps, rubella, oral polio, BCG,
             yellow fever, varicella, and TY21a typhoid vaccines.

          -  Patient is currently receiving or has received systemic corticosteroids ≤2 weeks prior
             to starting study drug, or who have not fully recovered from side effects of such
             treatment.

             • The following uses of corticosteroids are permitted: single doses, topical
             applications (e.g., for rash), inhaled sprays (e.g., for obstructive airways
             diseases), eye drops or local injections (e.g., intra-articular)

          -  Patient is currently receiving warfarin or other coumarin-derived anticoagulant for
             treatment, prophylaxis or otherwise. Therapy with heparin, low molecular weight
             heparin (LMWH) or fondaparinux is allowed.

          -  Treatment with an investigational agent within 30 days prior or within 5 half-lives
             (whichever is longer) before the first dose of the study drug.

          -  Patient who has received radiotherapy ≤4 weeks or limited field radiation for
             palliation ≤2 weeks prior to starting study drug, and who has not recovered to Grade 1
             or better from related side effects of such therapy (exceptions include alopecia)
             and/or in whom ≥25% of the bone marrow (Ellis, 1961) was irradiated.

          -  Patient has had major surgery within 14 days prior to starting study drug or has not
             recovered from major side effects (tumor biopsy is not considered as major surgery).

          -  Patient has not recovered from all toxicities related to prior anticancer therapies to
             NCI-CTCAE version 4.03 Grade ≤1 (Exception to this criterion: patients with any Grade
             of alopecia are allowed to enter the study).

          -  Patient with a Child-Pugh score B or C. See Appendix A.

          -  Patient has a history of non-compliance to medical regimen or inability to grant
             consent.

          -  Pregnant or nursing (lactating) women, where pregnancy is defined as the state of a
             female after conception and until the termination of gestation, confirmed by a
             positive hCG laboratory test.

          -  Women of child-bearing potential, defined as all women physiologically capable of
             becoming pregnant, unless they are using highly effective methods of contraception
             throughout the study and for at least 3 months after completion of treatment. Highly
             effective contraception methods include:

               -  Total abstinence when this is in line with the preferred and usual lifestyle of
                  the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal,
                  post-ovulation methods) and withdrawal are not acceptable methods of
                  contraception

               -  Female sterilization (have had surgical bilateral oophorectomy with or without
                  hysterectomy), total hysterectomy, or tubal ligation at least six weeks before
                  taking study treatment. In case of oophorectomy alone, only when the reproductive
                  status of the woman has been confirmed by follow up hormone level assessment

               -  Male sterilization (at least 6 months prior to screening). The vasectomized male
                  partner should be the sole partner for that patient

               -  Use of 1 highly effective method of contraception, and 1 additional (barrier)
                  method, at the same time

        Highly effective methods: Intra-uterine devices (IUD), Hormonal (birth control pills/oral
        contraceptives, injectable contraceptives, contraceptive patches, or contraceptive
        implants) Other effective methods (barrier methods): Latex condom, Diaphragm with
        spermicide; Cervical cap; Sponge

          -  Women are considered post-menopausal and not of child bearing potential if they have
             had 12 months of natural (spontaneous) amenorrhea with an appropriate clinical profile
             (e.g. age appropriate, history of vasomotor symptoms) or have had surgical bilateral
             oophorectomy (with or without hysterectomy) or tubal ligation at least six weeks ago.
             In the case of oophorectomy alone, only when the reproductive status of the woman has
             been confirmed by follow up hormone level assessment is she considered not of child
             bearing potential.

          -  Sexually active males unless they use a condom during intercourse while taking the
             drug and for 21 days after stopping treatment and should not father a child in this
             period. A condom is required to be used also by vasectomized men in order to prevent
             delivery of the drug via seminal fluid.

        Inclusion Criteria Step 2 (LMS patients only):

        - Patients with leiomyosarcoma must have tumors with intact Rb as documented by protein
        expression by IHC for study entry. Patients without sufficient archival tissue for testing
        will not be eligible. Please see Section 9.0 for further details on testing and
        interpretation of results. In the event that a patient has prior sequencing information
        (i.e. through commercial testing) suggestive of intact Rb, the patient may be included into
        the study on a case by case basis as determined by the principle investigators. The patient
        will still be required to submit tissue for Rb determination by IHC, but will not need to
        wait for these results for study entry.