The paper deals with moduli of continuity for paths of random processes indexed by a general metric space $Θ$ with values in a general metric space $\mathcal{X}$. Adapting the moment condition on the increments from the classical Kolmogorov-Chentsov theorem, the obtained result on the modulus of continuity allows for Hölder-continuous modifications if the metric space $\mathcal{X}$ is complete. This result is universal in the sense that its applicability depends only on the geometry of the space $Θ$. In particular, it is always applicable if $Θ$ is a bounded subset of a Euclidean space or a relatively compact subset of a connected Riemannian manifold. The derivation is based on refined chaining techniques developed by Talagrand. As a consequence of the main result a criterion is presented to guarantee uniform tightness of random processes with continuous paths.This is applied to find central limit theorems for Banach-valued random processes.