  In this paper we discuss the continuity properties of the integrated density
of states for random models based on that of the single site distribution. Our
results are valid for models with independent randomness with arbitrary free
parts. In particular in the case of the Anderson type models (with stationary,
growing, decaying randomness) on the $\nu$ dimensional lattice, with or without
periodic and almost periodic backgrounds, we show that if the single site
distribution is uniformly $\alpha$-H\"older continuous, $ 0 < \alpha \leq 1$,
then the density of states is also uniformly $\alpha$-H\"older continuous.
