Statistical models of unobserved heterogeneity are typically formalized as mix- tures of simple parametric models and interest naturally focuses on testing for homogeneity versus general mixture alternatives. Many tests of this type can be interpreted as C(») tests, as in Neyman (1959), and shown to be locally, asymptotically optimal. A unified approach to analysing the asymptotic behavior of such tests will be described, employing a variant of the LeCam LAN framework. These C(») tests will be contrasted with a new approach to likelihood ratio testing for mixture models. The latter tests are based on esti- mation of general (nonparametric) mixture models using the Kiefer and Wolfowitz (1956) maximum likelihood method. Recent developments in convex optimization are shown to dramatically improve upon earlier EM methods for computation of these estimators, and new results on the large sample behavior of likelihood ratios involving such estimators yield a tractable form of asymptotic inference. We compare performance of the two approaches identifying circumstances in which each is preferred.