Optimal stopping of a sequence of random variables is studied, with emphasis on generalized objectives which may be non-monotone functions of EXt , where t is a stopping time, or may even depend on the entire vector (E[X 1 I {t=l}], · ··, E[XnI {t=n}]), such as the minimax objective to maximize minj {E[XjI {t=j}]}. Convexity is used to establish a prophet inequality and universal bounds for the optimal return, and a method for constructing optimal stopping times for such objectives is given.