We distinguish two fundamentally different types of superluminal (group velocity>c) behavior. One type, called causal, is shown not to conflict with the usual condition that a signal originating in a region is felt only in the forward light cone of that region. This type of mode occurs as small oscillations about an unstable equilibrium configuration of high energy density. The second type of superluminosity, called noncausal, violates the causality conditions of conventional theory. It is shown that the noncausal superluminosity occurs only for field theories which are singular in the sense that there does not exist a unique one-to-one relationship between the momenta and the velocities. The field equations of such theories do not present a good Cauchy problem for arbitrary data on spacelike hyperplanes. They thus lead to ambiguous and/or singular solutions. An interesting feature of our first model is that it leads to tachyonlike behavior without one's having to introduce negative energies.