  In this paper we analyze some aspects of {\em exponential flights}, a
stochastic process that governs the evolution of many random transport
phenomena, such as neutron propagation, chemical/biological species migration,
or electron motion. We introduce a general framework for $d$-dimensional
setups, and emphasize that exponential flights represent a deceivingly simple
system, where in most cases closed-form formulas can hardly be obtained. We
derive a number of novel exact (where possible) or asymptotic results, among
which the stationary probability density for 2d systems, a long-standing issue
in Physics, and the mean residence time in a given volume. Bounded or
unbounded, as well as scattering or absorbing domains are examined, and Monte
Carlo simulations are performed so as to support our findings.
