We consider the integral functional $\int f\left(x,Du\right)dx$ under non standard growth assumptions of $\left(p,q\right)$-type: namely, we assume that ${\left|z\right|}^{p\left(x\right)}\le f\left(x,z\right)\le L\left(1+{\left|z\right|}^{p\left(x\right)}\right)$, a relevant model case being the functional $\int {\left|Du\right|}^{p\left(x\right)}dx$. Under sharp assumptions on the continuous function $p\left(x\right)>1$ we prove regularity of minimizers both in the scalar and in the vectorial case, in which we allow for quasiconvex energy densities. Energies exhibiting this growth appear in several models from mathematical physics.