In condensed matter physics, the term "chiral anomaly" implies the violation of the separate number conservation laws of Weyl fermions of different chiralities in the presence of parallel electric and magnetic fields. One effect of the chiral anomaly in the recently discovered Dirac and Weyl semimetals is a positive longitudinal magnetoconductance. Here we show that chiral anomaly and nontrivial Berry curvature effects engender another striking effect in Weyl semimetals, the planar Hall effect (PHE). Remarkably, the PHE manifests itself when the applied current, magnetic field, and the induced transverse "Hall" voltage all lie in the same plane, precisely in a configuration in which the conventional Hall effect vanishes. In this work we treat the PHE quasiclassically, and predict specific experimental signatures for type-I and type-II Weyl semimetals that can be directly checked in experiments.