This is a multi-center study to assess the safety and to determine the maximum tolerated dose of the combination of imatinib and LBH589 in patients with newly diagnosed and recurrent chordoma. For the recurrent population, those patients that do not require immediate surgical resection will be eligible. Patients will be treated with 4 cycles, followed by surgical resection if possible. If indicated, surgery may take place prior to the completion of 4 cycles.