This is a 3-part (Part A, Part B, Part C), Phase I/IIa, multi-center, open label, study in pediatric subjects with refractory or recurrent tumors. Part A is a repeat dose, dose escalation monotherapy study that will identify the recommended phase II dose (RP2D) on the continuous dosing schedule using a 3 + 3 dose-escalation procedure. Part B will evaluate the preliminary activity of trametinib monotherapy in 4 disease-specific cohorts of subjects. Each cohort will enroll at least 10 response-evaluable subjects (evaluable for response is defined as a subject with a pre-dose and at least 1 post-dose disease assessment or clinical assessment of progression of disease). The specifics around the Part C will be determined after completion of enrollment into Part A. The overall goal of this trial is to efficiently establish safe, pharmacologically relevant dose of trametinib in infants, children and adolescents and determine preliminary activity of trametinib monotherapy in selected recurrent, refractory or unresectable childhood tumors.