Inclusion Criteria:

* DISEASE RELATED CRITERIA
* Patients must have histologically or cytologically documented carcinoma primary to the intra- or extra-hepatic biliary system or gall bladder with clinical and/or radiologic evidence of unresectable, locally advanced or metastatic disease; patients with ampullary carcinoma are not eligible
* Patients must have measurable disease; computed tomography (CT) scans or magnetic resonance imaging (MRI)s used to assess measurable disease must have been completed within 28 days prior to registration; CT scans or MRIs used to assess non-measurable disease must have been completed within 42 days prior to registration; all disease must be assessed and documented on the Baseline Tumor Assessment Form (Response Evaluation Criteria in Solid Tumors \[RECIST\] 1.1)
* PRIOR/CONCURRENT THERAPY CRITERIA
* Patients must have completed any prior chemotherapy at least 21 days prior to registration and have recovered from any of the effects AND

* Patients must have experienced progression to no more than 1 prior regimen of systemic chemotherapy for advanced biliary cancer OR
* Patients who received adjuvant chemotherapy and had evidence of disease recurrence within 6 months of completion of the adjuvant treatment are also eligible; if patient received adjuvant treatment and had disease recurrence after 6 months, patients will only be eligible after failing one regimen of systemic chemotherapy used to treat the (unresectable or metastatic) disease recurrence
* Patients must not have been treated with prior MEK inhibitors; prior 5-FU or capecitabine treatment is allowed only if given as a radiosensitizer concurrently with radiation therapy at least 12 weeks prior to registration or if given as part of any adjuvant therapy regimen >= 12 months prior to study enrollment
* Patients must have no plans to receive concurrent chemotherapy, hormonal therapy, radiotherapy, immunotherapy or any other type of therapy (including herbal or natural supplements) for treatment of cancer while on this treatment protocol
* For patients who have received prior cryotherapy, radiation therapy, radiofrequency ablation, therasphere, ethanol injection, transarterial chemoembolization (TACE) or photodynamic therapy, the following criteria must be met:

* 28 days have elapsed since that therapy (lesions that have not been treated with local therapy must be present and measureable)
* CLINICAL/LABORATORY CRITERIA
* Patients must have a Zubrod performance status of 0-1
* Absolute neutrophil count (ANC) > 1000/mcL
* Platelets > 100000/mcL
* Total bilirubin =< 2.0 x the institutional upper limit of normal limits (IULN)
* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) both =< 3 x IULN; if liver metastases are present, AST and ALT must be =< 5 x IULN
* If the patient has had decompression of the biliary tree within the last 14 days, stability of the bilirubin level needs to be confirmed with two measurements that are within 5 to 7 days of each other; (the second measurement must be obtained within 7 days prior to registration;) both the first and second measurement must be =< 2.0 x IULN; stability is defined as the second measurement being no more than one point higher than the first
* Patients must have adequate kidney function as evidenced by at least ONE of the following:

* Serum creatinine =< 1.5 x IULN within 28 days prior to registration
* Calculated creatinine clearance >= 50 ml/min for patients with creatinine level of 1.0-1.5 x IULN; the serum creatinine value used in the calculation must have been obtained within 28 days prior to registration
* Patients with known history or current evidence of retinal vein occlusion (RVO) or retinal pigment epithelial detachment (RPED) are not eligible:

* History of RVO or RPED, or predisposing factors to RVO or RPED (e.g. such as uncontrolled glaucoma or ocular hypertension, uncontrolled systemic disease such as hypertension, diabetes mellitus, or history of hyperviscosity or hypercoagulability syndromes)
* Visible retinal pathology as assessed by ophthalmic exam that is considered a risk factor for RVO or RPED such as:

* Evidence of new optic disc cupping
* Evidence of new visual field defects
* Intraocular pressure > 21 mmHg
* NOTE: ophthalmic exam is required for all patients; this exam should not be performed until or unless it is very clear the patient is otherwise eligible for registration; this exam should not be performed until or unless it is very clear the patient is otherwise eligible for registration
* Patients must have echocardiogram and left ventricular ejection fraction (LVEF) >= institutional lower limit of normal (LLN) within 28 days prior to registration; this exam should not be performed until or unless it is very clear the patient is otherwise eligible for registration
* Patients must not have uncontrolled or clinically significant cardiovascular disease including: myocardial infarction within past 6 months; uncontrolled angina within past 6 months; class II-IV New York Heart Association (NYHA) congestive heart failure; grade 3 cardiac valve dysfunction; cardiac arrhythmia not controlled by medication; history of stroke or transient ischemic attack within 6 months; history of arterial thrombotic event (ATE) of any type in the past 6 months; treatment-refractory hypertension defined as a blood pressure of systolic > 140 mmHg and/or diastolic > 90 mmHg which cannot be controlled with anti-hypertensive therapy; known intra-cardiac defibrillators; known cardiac metastases
* Patients must have an electrocardiogram (ECG) within 28 days prior to registration; patients must have corrected QT interval (QTc) =< 500 msec; this exam should not be performed until or unless it is very clear the patient is otherwise eligible for registration
* Must be 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
* Must not have a known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to trametinib, or excipients or to dimethyl sulfoxide (DMSO) or other agents used in study
* Must not have active hepatitis B virus (HBV), or hepatitis C virus (HCV) infection (patients with chronic or cleared HBV and HCV infection are eligible)
* Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible
* Patients must not be pregnant or nursing; women/men of reproductive potential must have agreed to use an effective contraceptive method while on study and for 4 months after discontinuation of study drug; a woman is considered to be of "reproductive potential" if she has had menses at any time in the preceding 12 consecutive months; in addition to routine contraceptive methods, "effective contraception" also includes heterosexual celibacy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) defined as a hysterectomy, bilateral oophorectomy or bilateral tubal ligation; however, if at any point a previously celibate patient chooses to become heterosexually active during the time period for use of contraceptive measures outlined in the protocol, he/she is responsible for beginning contraceptive measures
* No other prior malignancy is allowed except for the following: adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, adequately treated stage I or II cancer from which the patient is currently in complete remission, or any other cancer from which the patient has been disease free for five years
* SPECIMEN SUBMISSION CRITERIA
* Patients must submit paraffin-embedded tissue and blood for banking within 28 days after registration; paraffin-embedded tissue from prior surgical resection or from a diagnostic biopsy is acceptable
* REGULATORY CRITERIA
* Patients or their legally authorized representative must be informed of the investigational nature of this study and must sign and give written informed consent in accordance with institutional and federal guidelines
* As a part of the Oncology Patient Enrollment Network (OPEN) registration process the treating institution's identity is provided in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered in the system