Inclusion Criteria:

          -  Patients must have a histologically or cytologically confirmed solid malignancy that
             is metastatic or unresectable for which standard curative or palliative treatments do
             not exist or are no longer effective

               -  Hepatocellular carcinoma (HCC) patients are not required to have histologically
                  or cytologically confirmed malignancy, patients are considered eligible based on
                  tumor markers and/or imaging assessment

               -  Based on recent data that have shown limited trametinib benefit, patients with
                  the following tumor types will be excluded from the normal and mild cohorts:

                    -  Pancreatic cancer patients

                    -  Colorectal cancer patients

                    -  BRAF V600E melanoma patients who have failed BRAF inhibitors

                         -  Note: Patients with pancreatic cancer, colorectal cancer, and BRAF
                            V600E melanoma patients who have failed BRAF inhibitors are allowed to
                            enroll in the moderate and severe cohorts provided the patients: 1)
                            sign a separate consent form which outlines the extremely limited
                            activity observed in prior studies, and 2) are consented to the study
                            by a protocol-specified designee who is not their longitudinal
                            oncologist

          -  All patients must have completed any prior chemotherapy, targeted therapy,
             radiotherapy (unless palliative doses which must be discussed with study principal
             investigator), surgery, anti-angiogenic therapy or interferon >= 28 days before study
             entry

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 2 (Karnofsky >= 60%)

          -  Life expectancy of greater than 3 months

          -  Able to swallow and retain orally-administered medication and does not have any
             clinically significant gastrointestinal abnormalities that may alter absorption such
             as malabsorption syndrome or major resection of the stomach or bowels

          -  All prior treatment-related toxicities must be Common Terminology Criteria for Adverse
             Events (CTCAE) version 4.0 grade =< 1 (except alopecia) at the time of enrollment

          -  Absolute neutrophil count (ANC) >= 1.2 x 10^9/L

          -  Hemoglobin >= 9 g/dL

          -  Platelets >= 75 x 10^9/L

          -  Serum creatinine =< 1.5 mg/dL (=< 133 umol/L) OR calculated creatinine clearance
             (Cockcroft-Gault formula) >= 50 mL/min OR 24-hour urine creatinine clearance >= 50
             mL/min

          -  Proteinuria =< +1 on dipstick or =< 1 gram/24 hours

          -  Prothrombin time (PT) =< 1.5 x institutional upper limit of normal (ULN)

          -  International normalized ratio (INR) =< 1.5 x institutional ULN

          -  Partial thromboplastin time (PTT) =< 1.5 x institutional ULN

          -  Left ventricular ejection fraction (LVEF) >= institutional lower limit of normal (LLN)
             by echocardiogram (ECHO) or multigated acquisition scan (MUGA)

          -  No distinction should be made between liver dysfunction due to metastases and liver
             dysfunction due to other causes

          -  Patients with abnormal hepatic function will be eligible and will be grouped according
             to criteria summarized below:

               -  Group A: Normal hepatic function

                    -  Bilirubin =< ULN

                    -  Aspartate aminotransferase (AST) =< ULN

               -  Group B: Mild hepatic dysfunction

                    -  B1: bilirubin =< ULN and AST > ULN

                    -  B2: ULN < bilirubin =< 1.5 x ULN and any AST

               -  Group C: Moderate hepatic dysfunction

                    -  1.5 x ULN < bilirubin =< 3 x ULN and any AST

               -  Group D: Severe hepatic dysfunction

                    -  3 x ULN < bilirubin =< 10 x ULN and any AST; hepatic function tests should
                       be repeated within 24 hours prior to starting initial therapy and may result
                       in patients' group assignment being altered if different to registration
                       test results

               -  Patients with active hemolysis should be excluded. No distinction should be made
                  between liver dysfunction due to metastases and liver dysfunction due to other
                  causes. Hepatic function tests should be repeated within 24 hours prior to
                  starting initial therapy and may result in patients' group assignment being
                  altered if different to registration test results

          -  Trametinib can cause fetal harm when administered to a pregnant woman; women of
             child-bearing potential and men must agree to use adequate contraception (hormonal or
             barrier method of birth control; abstinence) prior to study entry, during the study
             participation, and for four months after the last dose of the drug; women of
             child-bearing potential must have a negative serum pregnancy test within 14 days prior
             to registration and agree to use effective contraception throughout the treatment
             period and for 4 months after the last dose of study treatment; should a woman become
             pregnant or suspect she is pregnant while she or her partner is participating in this
             study, she should inform her treating physician immediately

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

        Exclusion Criteria:

          -  History of another malignancy

               -  Exception: patients who have been disease-free for 3 years, or patients with a
                  history of completely resected non-melanoma skin cancer and/or patients with
                  indolent secondary malignancies, are eligible; consult the Cancer Therapy
                  Evaluation Program (CTEP) Medical Monitor if unsure whether second malignancies
                  meet the requirements specified above

          -  History of interstitial lung disease or pneumonitis

          -  Any major surgery, extensive radiotherapy, chemotherapy with delayed toxicity,
             biologic therapy, or immunotherapy within 28 days prior to enrollment and/or daily or
             weekly chemotherapy without the potential for delayed toxicity within 14 days prior to
             enrollment

          -  Use of other investigational drugs within 28 days (or five half-lives, whichever is
             shorter; with a minimum of 14 days from the last dose) preceding the first dose of
             trametinib and during the study; patients previously treated with v-raf murine sarcoma
             (RAF) and/or mitogen-activated protein kinase (MEK) inhibitors are excluded from the
             study; multikinase antiangiogenic tyrosine kinase inhibitors such as regorafenib,
             sorafenib, sunitinib, etc. whose primary mechanism of action is not RAF inhibition,
             are allowed; if there are any questions, please contact study's principal investigator

          -  Symptomatic or untreated leptomeningeal or brain metastases or spinal cord compression

          -  Have a known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs
             chemically related to trametinib or excipients or to dimethyl sulfoxide (DMSO)

          -  Current use of a prohibited medication; the following medications or non-drug
             therapies are prohibited:

               -  Other anti-cancer therapy while on study treatment; (Note: megestrol [Megace] if
                  used as an appetite stimulant is allowed)

               -  Concurrent treatment with bisphosphonates is permitted; however, treatment must
                  be initiated prior to the first dose of study therapy; prophylactic use of
                  bisphosphonates in patients without bone disease is not permitted, except for the
                  treatment of osteoporosis

               -  Because the composition, PK, and metabolism of many herbal supplements are
                  unknown, the concurrent use of all herbal supplements is prohibited during the
                  study (including, but not limited to, St. John's wort, kava, ephedra [ma huang],
                  ginkgo biloba, dehydroepiandrosterone [DHEA], yohimbe, saw palmetto, or ginseng)

          -  History or current evidence/risk of retinal vein occlusion (RVO)

          -  History or evidence of cardiovascular risk including any of the following:

               -  LVEF < LLN

               -  A QT interval corrected for heart rate using the Bazett's formula QTcB >= 480
                  msec

               -  History or evidence of current clinically significant uncontrolled arrhythmias
                  (exception: patients with controlled atrial fibrillation for > 30 days prior to
                  randomization are eligible)

               -  History of acute coronary syndromes (including myocardial infarction and unstable
                  angina), coronary angioplasty, or stenting within 6 months prior to randomization

               -  History or evidence of current >= class II congestive heart failure as defined by
                  the New York Heart Association (NYHA) functional classification system

               -  Treatment-refractory hypertension defined as a blood pressure of systolic > 140
                  mmHg and/or diastolic > 90 mmHg which cannot be controlled by anti-hypertensive
                  therapy

               -  Patients with intra-cardiac defibrillators

               -  Known cardiac metastases

          -  Active hepatitis B virus (HBV), or hepatitis C virus (HCV) infection (patients with
             chronic or cleared HBV and HCV infection are eligible)

          -  Patients with known human immunodeficiency virus (HIV) infection are eligible if not
             on antiviral agents and cluster of differentiation (CD)4 counts are adequate (>= 500)

          -  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

          -  Animal reproductive studies have not been conducted with trametinib; therefore, the
             study drug must not be administered to pregnant women or nursing mothers; women of
             childbearing potential should be advised to avoid pregnancy and use effective methods
             of contraception; men with a female partner of childbearing potential must have either
             had a prior vasectomy or agree to use effective contraception; if a female patient or
             a female partner of a patient becomes pregnant while the patient receives trametinib,
             the potential hazard to the fetus should be explained to the patient and partner (as
             applicable)

          -  HIV-positive patients on combination antiretroviral therapy are ineligible because of
             the potential for pharmacokinetic interactions; in addition, these patients are at
             increased risk of lethal infections when treated with marrow-suppressive therapy;
             appropriate studies will be undertaken in patients receiving combination
             antiretroviral therapy when indicated

          -  Any condition or medical problem in addition to the underlying malignancy and organ
             dysfunction which the investigator feels would pose unacceptable risk