  Current fluctuations are a powerful tool to unravel the underlying physics of
the observed transport process. This work discusses some general properties of
the third and the fourth current cumulant (skewness and kurtosis) related to
dynamics and thermodynamics of a transport setup. Specifically, several
distinct bounds on these quantities are either analytically derived or
numerically conjectured, which are applicable to: 1) noninteracting fermionic
systems, 2) noninteracting bosonic systems, 3) thermally driven classical
Markovian systems, 4) unicyclic Markovian networks. Finally, it is demonstrated
that violation of the obtained inequalities can provide a broad spectrum of
information about the physics of the analyzed system, e.g., enable one to infer
the presence of interactions or unitary dynamics, unravel the topology of the
Markovian network, or characterize the nature of thermodynamic forces driving
the system. In particular, relevant information about the microscopic dynamics
can be gained even at equilibrium when the current variance -- a standard
measure of current fluctuations -- is determined mostly by the thermal noise.
