  We explore the problem of binary classification in machine learning, with a
twist - the classifier is allowed to abstain on any datum, professing ignorance
about the true class label without committing to any prediction. This is
directly motivated by applications like medical diagnosis and fraud risk
assessment, in which incorrect predictions have potentially calamitous
consequences. We focus on a recent spate of theoretically driven work in this
area that characterizes how allowing abstentions can lead to fewer errors in
very general settings. Two areas are highlighted: the surprising possibility of
zero-error learning, and the fundamental tradeoff between predicting
sufficiently often and avoiding incorrect predictions. We review efficient
algorithms with provable guarantees for each of these areas. We also discuss
connections to other scenarios, notably active learning, as they suggest
promising directions of further inquiry in this emerging field.
