We prove that the Sherrington-Kirkpatrick model of spin glasses is chaotic under small perturbations of the couplings at any temperature in the absence of an external field. The result is proved for two kinds of perturbations: (a) distorting the couplings via Ornstein-Uhlenbeck flows, and (b) replacing a small fraction of the couplings by independent copies. We further prove that the S-K model exhibits multiple valleys in its energy landscape, in the weak sense that there are many states with near-minimal energy that are mutually nearly orthogonal. We show that the variance of the free energy of the S-K model is unusually small at any temperature. (By `unusually small' we mean that it is much smaller than the number of sites; in other words, it beats the classical Gaussian concentration inequality, a phenomenon that we call `superconcentration'.) We prove that the bond overlap in the Edwards-Anderson model of spin glasses is not chaotic under perturbations of the couplings, even large perturbations. Lastly, we obtain sharp lower bounds on the variance of the free energy in the E-A model on any bounded degree graph, generalizing a result of Wehr and Aizenman and establishing the absence of superconcentration in this class of models. Our techniques apply for the p-spin models and the Random Field Ising Model as well, although we do not work out the details in these cases.