This is an open-label, multicenter, dose escalation, phase 1 study of MLN8237 in adult patients with advanced solid tumors. This study will be the first to administer MLN8237 to humans. MLN8237 will be given daily for 7 to 21 consecutive days followed by a 14-day recovery period (cycle length = 21 to 35 days). Patients will continue to receive repeated cycles of MLN8237 as long as their disease has not progressed and they have not experienced unacceptable MLN8237-related toxicity. The dose of MLN8237 will be increased in successive cohorts of 3 to 6 patients until a maximum tolerated dose of MLN8237 has been identified. Once a maximum tolerated dose has been identified for 7 consecutive days of treatment, a total of 9 to 12 patients will be treated at that dose to better define its safety, PK, and pharmacodynamics. Longer periods of dosing with MLN8237 then will be evaluated, with an increase from 7 to 14 days, and, if feasible, to 21 days. Intermediate lengths of dosing may also be evaluated. A 14-day recovery period will follow each dosing period regardless of its duration.