  The paper deals with moduli of continuity for paths of random processes
indexed by a general metric space $\Theta$ 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 H\"older-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 $\Theta$. In
particular, it is always applicable if $\Theta$ 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.
