  In this article we study the existence of limit cycles in families of
piecewise smooth differential equations having the unit circle as discontinuity
region. We consider families presenting singularities of center or saddle type,
visible or invisible, as well as the case without singularities. We establish
an upper bound for the number of limit cycles and give examples showing that
the maximum number of limit cycles can be reached. We also discuss the
existence of homoclinic cycles for such differential equations in the
saddle-center case.
