Inclusion Criteria:

          -  Patients must have measurable disease as defined by Response Evaluation Criteria in
             Solid Tumors (RECIST) version 1.1

          -  Eligible patients should have a histologically confirmed locally advance or
             metastatic TCC who progressed after standard therapy

          -  Patient must have TCCs tumors harboring a TSC1 or TSC2 mutation identified by a
             Clinical Laboratory Improvement Amendments (CLIA) certified laboratory

          -  Patients must have TCC tumor tissue available for submission in a form of at least 10
             unstained slides or formalin-fixed paraffin-embedded (FFPE) block (FFPE block highly
             recommended and preferred); if the tissue were previously sent to Yale Tumor
             Profiling Laboratory (TPL) for prescreening, additional tissue submission may not be
             required if sufficient remaining tissue is available; but, must first consult with
             the principle investigator

          -  Patient must have developed disease progression during or following treatment with at
             least one platinum- containing regimen (e.g., gemcitabine/cisplatin [GC],
             methotrexate-vinblastine-doxorubicin-cisplatin [MVAC], carboplatin, gemcitabine
             [CarboGem]) for inoperable locally advanced or metastatic urothelial carcinoma or
             disease recurrence, or must be unfit or ineligible for cisplatin-based chemotherapy;
             there is no restriction on the number of prior lines of chemotherapeutics agents
             received

               -  Patients who progressed within 12 months of treatment with a platinum-containing
                  neoadjuvant or adjuvant regimen are considered second-line patients; therefore,
                  these patients may be also eligible

               -  Patients who are unfit or ineligible for cisplatin-based chemotherapy as defined
                  by any one of the following criteria are eligible for this trial:

                    -  Eastern Cooperative Oncology Group (ECOG) performance score of 2

                    -  Creatinine clearance < 60 mL/min

                    -  A hearing loss (measured by audiometry) of 25 dB at two contiguous
                       frequencies

                    -  Grade >= 2 peripheral neuropathy

          -  ECOG performance status =< 2 (Karnofsky >= 60 %)

          -  Life expectancy of greater than 12 weeks

          -  Hemoglobin >= 9 g/dL

          -  Fasting serum glucose =< 130 mg/dL

          -  Glycosylated hemoglobin measurement (HbA1c) < 7.0%

          -  Fasting triglycerides =< 300 mg/dL

          -  Leukocytes >= 3,000/mcL

          -  Absolute neutrophil count >= 1,500/mcL

          -  Platelets >= 100,000/mcL

          -  Total bilirubin within normal institutional limits

          -  Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT])
             =< 2.5 × institutional upper limit of normal (ULN) and =< 5 ULN if liver metastases
             are present

          -  Creatinine within normal institutional limits OR creatinine clearance >= 50 mL/min
             based either on Cockroft-Gault estimate or based on urine collection (12 or 24 hour)

          -  Patients with controlled diabetes are allowed on study; controlled diabetes is
             defined as fasting blood sugar (FBS) < 130 mg/dL, and patients whose FBS can be
             brought in this range with medical therapy are eligible for trial inclusion

          -  Women of childbearing age should avoid becoming pregnant while taking any mTOR
             inhibitor including MLN0128 (TAK-228)

               -  Female patients must:

                    -  Be postmenopausal for at least 1 year before the screening visit, OR

                    -  Be surgically sterile, OR

                    -  If they are of childbearing potential, agree to practice 1 highly effective
                       method of contraception and 1 additional effective (barrier) method, at the
                       same time, from the time of signing the informed consent through 90 days
                       (or longer, as mandated by local labeling [e.g., USPI, SmPC, etc.;]) after
                       the last dose of study drug, OR

                    -  Agree to practice true abstinence, when this is in line with the preferred
                       and usual lifestyle of the patient; NOTE: periodic abstinence [e.g.,
                       calendar, ovulation, symptothermal, postovulation methods], withdrawal,
                       spermicides only, and lactational amenorrhea are not acceptable methods of
                       contraception; female and male condoms should not be used together

               -  Male patients, even if surgically sterilized (i.e., status postvasectomy), must:

                    -  Agree to practice highly effective barrier contraception during the entire
                       study treatment period and through 120 days after the last dose of study
                       drug, OR

                    -  Agree to practice true abstinence, when this is in line with the preferred
                       and usual lifestyle of the patient (NOTE: periodic abstinence [e.g.,
                       calendar, ovulation, symptothermal, postovulation methods for the female
                       partner], withdrawal, spermicides only, and lactational amenorrhea are not
                       acceptable methods of contraception; female and male condoms should not be
                       used together)

                    -  AND agree not to donate sperm during the course of this study or within 120
                       days after receiving their last dose of study drug

          -  Ability to swallow oral medications

          -  Ability to understand and the willingness to sign a written informed consent document

        Exclusion Criteria:

          -  Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for
             nitrosoureas or mitomycin C) prior to entering the study or those who have not
             recovered from adverse events due to agents administered more than 4 weeks earlier

          -  Patients who are receiving any other investigational agents

          -  Patients with known untreated brain metastases should be excluded from this clinical
             trial

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to MLN0128 (TAK-228)

          -  Subjects who are on systemic corticosteroids (intravenous (IV) or oral steroids,
             excluding inhaled, topical or ophthalmic corticosteroids), or anti-epileptic drugs
             for treated brain metastasis

          -  Subjects taking strong inhibitors and/or inducers of cytochrome P450 (CYP) 3A4,
             CYP2C19 or CYP2C9 within 1 week preceding the first dose of MLN0128 (TAK-228); if a
             subject requires treatment with strong inhibitors and/or inducers of CYP3A4, CYP2C19
             and/or CYP2C9, alternative treatment must be considered; if no alternative is
             available, one such medication may be allowed after discussing with the study
             principle investigator

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac
             arrhythmia, or psychiatric illness/social situations that would limit compliance with
             study requirements

          -  Pregnant and breastfeeding women are excluded from this study; breastfeeding should
             be discontinued if the mother is treated with MLN0128 (TAK-228)

          -  Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral
             therapy are ineligible; however, HIV patients treated with regimens that have low
             cytochrome P450 (CYP450) inhibition may be allowed as long as the patient's general
             health and cluster of differentiation (CD)4 counts are within acceptable levels

          -  Patients with symptomatic/untreated central nervous system (CNS) metastases; patients
             with treated and stable brain metastasis are allowed

          -  Patients with impaired cardiac function or clinically significant cardiac disease;
             patients with baseline prolongation of the rate-corrected QT interval (QTc) (e.g.,
             repeated demonstration of QTc interval > 480 milliseconds, or history of congenital
             long QT syndrome, or torsades de pointes) will not be allowed; no ischemic myocardial
             or cerebrovascular event, class III or IV heart failure, placement of pacemaker, or
             pulmonary embolism within six months of receiving first dose of MLN0128 (TAK-228)

          -  Patients with untreated or active hepatitis B or C infection

          -  Significant active cardiovascular or pulmonary disease at the time of study entry,
             including

               -  Uncontrolled high blood pressure (i.e., systolic blood pressure > 180 mm Hg,
                  diastolic blood pressure > 95 mm Hg)

               -  Pulmonary hypertension

               -  Uncontrolled asthma or oxygen (O2) saturation < 90% by ABG (arterial blood gas)
                  analysis or pulse oximetry on room air

               -  Significant valvular disease; severe regurgitation or stenosis by imaging
                  independent of symptom control with medical intervention, or history of valve
                  replacement

               -  Medically significant (symptomatic) bradycardia

          -  Manifestations of malabsorption due to prior gastrointestinal (GI) surgery, GI
             disease, or for an unknown reason that may alter the absorption of MLN0128 (TAK-228)

          -  Concomitant administration of any proton pump inhibitor (PPI) is not permitted during
             the study; patients receiving PPI therapy before enrollment must stop using the PPI
             for 7 days before their first dose of study drugs

          -  History of any of the following within the last 6 months prior to study entry:

               -  Ischemic myocardial event, including angina requiring therapy and artery
                  revascularization procedures

               -  Ischemic cerebrovascular event, including transient ischemic attack (TIA) and
                  artery revascularization procedures

               -  Requirement for inotropic support (excluding digoxin) or serious (uncontrolled)
                  cardiac arrhythmia (including atrial flutter/fibrillation, ventricular
                  fibrillation or ventricular tachycardia)

               -  Pulmonary embolism

          -  Subjects who have initiated treatment with bisphosphonates less than 30 days prior to
             the first administration of MLN0128 (TAK-228); concurrent bisphosphonate use is only
             allowed if the bisphosphonate was initiated at least 30 days prior to the first
             administration of MLN0128 (TAK-228)

          -  Patients who received prior PI3K, AKT or mTOR inhibitors are not allowed

          -  Patients who received radiation therapy within the last 4 weeks; radiation exposure
             may not exceed 30% of marrow area