  A standard approach to analysis of noise-induced effects in stochastic
dynamics assumes a Gaussian character of the noise term describing interaction
of the analyzed system with its complex surroundings. An additional assumption
about the existence of timescale separation between the dynamics of the
measured observable and the typical timescale of the noise allows external
fluctuations to be modeled as temporally uncorrelated and therefore white.
However, in many natural phenomena the assumptions concerning the
abovementioned properties of "Gaussianity" and "whiteness" of the noise can be
violated. In this context, in contrast to the spatiotemporal coupling
characterizing general forms of non-Markovian or semi-Markovian L\'evy walks,
so called L\'evy flights correspond to the class of Markov processes which
still can be interpreted as white, but distributed according to a more general,
infinitely divisible, stable and non-Gaussian law. L\'evy noise-driven
non-equilibrium systems are known to manifest interesting physical properties
and have been addressed in various scenarios of physical transport exhibiting a
superdiffusive behavior. Here we present a brief overview of our recent
investigations aimed to understand features of stochastic dynamics under the
influence of L\'evy white noise perturbations. We find that the archetypal
phenomena of noise-induced ordering are robust and can be detected also in
systems driven by non-Gaussian, heavy-tailed fluctuations with infinite
variance.
