  We look into the problem of stochastic resetting with refractory periods. The
model dynamics comprises diffusive and motionless phases. The diffusive phase
ends at random time instants, at which the system is reset to a given position
-- where the system remains at rest for a random time interval, termed the
refractory period. A pathway formulation is introduced to derive exact
analytical results for the relevant observables in a broad framework, with the
resetting time and the refractory period following arbitrary distributions. For
the paradigmatic case of Poissonian distributions of the resetting and
refractory times, in general with different characteristic rates, closed-form
expressions are obtained that successfully describe the relaxation to the
steady state. Finally, we focus on the single-target search problem, in which
the survival probability and the mean first passage time to the target can be
exactly computed. Therein, we also discuss optimal strategies, which show a
non-trivial dependence on the refractory period.
