  We consider regularity issues for minima of non-autonomous functionals in the
Calculus of Variations exhibiting non-uniform ellipticity features. We provide
a few sharp regularity results for local minimizers that also cover the case of
functionals with nearly linear growth. The analysis is carried out provided
certain necessary approximation-in-energy conditions are satisfied. These are
related to the occurrence of the so-called Lavrentiev phenomenon that that
non-autonomous functionals might exhibit, and which is a natural obstruction to
regularity. In the case of vector valued problems we concentrate on higher
gradient integrability of minima. Instead, in the scalar case, we prove local
Lipschitz estimates. We also present an approach via a variant of Moser's
iteration technique that allows to reduce the analysis of several non-uniformly
elliptic problems to that for uniformly elliptic ones.
