  Motivated by classical considerations from risk theory, we investigate
boundary crossing problems for refracted L\'evy processes. The latter is a
L\'evy process whose dynamics change by subtracting off a fixed linear drift
(of suitable size) whenever the aggregate process is above a pre-specified
level. More formally, whenever it exists, a refracted L\'evy process is
described by the unique strong solution to the stochastic differential equation
\[ \D U_t = - \delta \mathbf{1}_{\{U_t >b\}}\D t + \D X_t \] where $X=\{X_t
:t\geq 0\}$ is a L\'evy process with law $\mathbb{P}$ and $b, \delta\in
\mathbb{R}$ such that the resulting process $U$ may visit the half line
$(b,\infty)$ with positive probability. We consider in particular the case that
$X$ is spectrally negative and establish a suite of identities for the case of
one and two sided exit problems. All identities can be written in terms of the
$q$-scale function of the driving L\'evy process and its perturbed version
describing motion above the level $b$. We remark on a number of applications of
the obtained identities to (controlled) insurance risk processes.
