While treatment for melanoma has evolved in the last few years, advanced/metastatic melanoma remains an aggressive disease with poor prognosis. Trametinib and nab-paclitaxel have demonstrated single-agent activity in Phase III trials of patients with melanoma. The investigators propose to investigate this drug combination in patients with advanced unresectable or metastatic melanoma. This open-label Phase I study is designed to determine the maximum tolerated dose (MTD), recommended Phase II dose (RP2D), and safety profile of trametinib and nab-paclitaxel given in combination. Patients with melanoma and known BRAF mutation status will receive escalating doses of trametinib and nab-paclitaxel to determine the MTD. Once the MTD is determined an additional 6 patients will be treated at the MTD to establish the dose for future studies.