We introduce a criterion, resilience, which allows properties of a dataset\n(such as its mean or best low rank approximation) to be robustly computed, even\nin the presence of a large fraction of arbitrary additional data. Resilience is\na weaker condition than most other properties considered so far in the\nliterature, and yet enables robust estimation in a broader variety of settings.\nWe provide new information-theoretic results on robust distribution learning,\nrobust estimation of stochastic block models, and robust mean estimation under\nbounded $k$th moments. We also provide new algorithmic results on robust\ndistribution learning, as well as robust mean estimation in $\\ell_p$-norms.\nAmong our proof techniques is a method for pruning a high-dimensional\ndistribution with bounded $1$st moments to a stable "core" with bounded $2$nd\nmoments, which may be of independent interest.\n