  Stochastic processes play a key role for modeling a huge variety of transport
problems out of equilibrium, with manifold applications throughout the natural
and social sciences. To formulate models of stochastic dynamics the
conventional approach consists in superimposing random fluctuations on a
suitable deterministic evolution. These fluctuations are sampled from
probability distributions that are prescribed a priori, most commonly as
Gaussian or L\'evy. While these distributions are motivated by (generalised)
central limit theorems they are nevertheless \textit{unbounded}, meaning that
arbitrarily large fluctuations can be obtained with finite probability. This
property implies the violation of fundamental physical principles such as
special relativity and may yield divergencies for basic physical quantities
like energy. Here we solve the fundamental problem of unbounded random
fluctuations by constructing a comprehensive theoretical framework of
stochastic processes possessing physically realistic finite propagation
velocity. Our approach is motivated by the theory of L\'evy walks, which we
embed into an extension of conventional Poisson-Kac processes. The resulting
extended theory employs generalised transition rates to model subtle
microscopic dynamics, which reproduces non-trivial spatio-temporal correlations
on macroscopic scales. It thus enables the modelling of many different kinds of
dynamical features, as we demonstrate by three physically and biologically
motivated examples. The corresponding stochastic models capture the whole
spectrum of diffusive dynamics from normal to anomalous diffusion, including
the striking `Brownian yet non Gaussian' diffusion, and more sophisticated
phenomena such as senescence. Extended Poisson-Kac theory can therefore be used
to model a wide range of finite velocity dynamical phenomena that are observed
experimentally.
