We establish the sample complexity of Approximate Hypothesis Testing (AHT) where—unlike in classical hypothesis testing—we need only approximate the hypothesis governing the observed samples rather than recover it exactly. We show that the AHT sample complexity scales inversely with the multivariate Bhatthacharyya distance evaluated on a “maximally confusable” subset of hypotheses that is characterized by the chosen distance measure and approximation accuracy. Index terms—hypothesis testing, sample complexity, learning, Bhattacharyya distance, Hellinger distance.