  This paper deals with existence and robust stability of hybrid limit cycles
for a class of hybrid systems given by the combination of continuous dynamics
on a flow set and discrete dynamics on a jump set. For this purpose, the notion
of Zhukovskii stability, typically stated for continuous-time systems, is
extended to the hybrid systems. Necessary conditions, particularly, a condition
using a forward invariance notion, for existence of hybrid limit cycles are
first presented. In addition, a sufficient condition, related to Zhukovskii
stability, for the existence of (or lack of) hybrid limit cycles is
established. Furthermore, under mild assumptions, we show that asymptotic
stability of such hybrid limit cycles is not only equivalent to asymptotic
stability of a fixed point of the associated Poincar\'{e} map but also robust
to perturbations. Specifically, robustness to generic perturbations, which
capture state noise and unmodeled dynamics, and to inflations of the flow and
jump sets are established in terms of $\mathcal{KL}$ bounds. Furthermore,
results establishing relationships between the properties of a computed
Poincar\'{e} map, which is necessarily affected by computational error, and the
actual asymptotic stability properties of a hybrid limit cycle are proposed. In
particular, it is shown that asymptotic stability of the exact Poincar\'{e} map
is preserved when computed with enough precision. Several examples, including a
congestion control system and spiking neurons, are presented to illustrate the
notions and results throughout the paper.
