Forward invariance for hybrid dynamical systems modeled by differential and difference inclusions with state-depending conditions enabling flows and jumps is studied. Several notions of forward invariance are considered and sufficient conditions in terms of the objects defining the system are introduced. In particular, we study forward invariance notions that apply to systems with nonlinear dynamics for which not every solution is unique or may exist for arbitrary long hybrid time. Such behavior is very common in hybrid systems. Lyapunov-based conditions are also proposed for the estimation of invariant sets. Applications and examples are given to illustrate the results. In particular, the results are applied to the estimation of weakly forward invariant sets, which is an invariance property of interest when employing invariance principles to study convergence of solutions.
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