Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric Φ on L induces a Hilbert space structure on the space of global holomorphic sections with values in the kth tensor power of L. In this paper various convergence results are obtained for the corresponding Bergman kernels (i.e., orthogonal projection kernels). The convergence is studied in the large k limit and is expressed in terms of the equilibrium metric Φe associated to the fixed metric Φ, as well as in terms of the Monge-Ampere measure of the metric Φ itself on a certain support set. It is also shown that the equilibrium metric is C1,1 on the complement of the augmented base locus of L. For L ample these results give generalizations of well-known results concerning the case when the curvature of Φ is globally positive (then Φe = Φ). In general, the results can be seen as local metrized versions of Fujita's approximation theorem for the volume of L.