The study is designed to explore the safety, tolerability and pharmacokinetics of ARQ 621 and define a recommended phase 2 (RP2D)dose of ARQ 621.Treatment will be initiated at a dose level of 10 mg/m^2 IV infusion for an hour once weekly in 4-week (28 day) consecutive and continuous cycles. ARQ 621 should be infused IV over two hours at doses 200 mg/m^2 and higher (cohort 8 and above). All cycles of therapy will consist of the patient taking ARQ 621 intravenously once weekly for 4 weeks. Dose escalation will proceed initially by doubling (cohorts 2 and 3) and subsequently by a modified Fibonacci scheme. Dose escalations will be performed using 3-6 patient cohorts. In these cohorts, if a single dose limiting toxicity (DLT) is experienced among patients 1-3, the dose cohort will be expanded to six patients. The maximum tolerated dose (MTD) will be defined as the dose level at which no greater than 1/6 patients experiences a DLT. Once an MTD is identified, up to an additional 20 patients (with types of malignancy to be determined at a later date by study investigator and clarified by study amendment) may be treated at this MTD of ARQ 621. If an MTD is not identified in the initial 10 dosing cohorts, dose escalation will proceed in a manner to be defined by subsequent amendment with the purpose of determining a RP2D of ARQ 621.