  In this paper we provide necessary and sufficient conditions for strongly
group graded rings to be simple. For a strongly group graded ring $R =
\bigoplus_{g\in G} R_g$ the grading group $G$ acts, in a natural way, as
automorphisms of the commutant of the neutral component subring $R_e$ in $R$
and of the center of $R_e$. We show that if $R$ is a strongly $G$-graded ring
where $R_e$ is maximal commutative in $R$, then $R$ is a simple ring if and
only if $R_e$ is $G$-simple (i.e. there are no nontrivial $G$-invariant
ideals). We also show that if $R_e$ is commutative (not necessarily maximal
commutative) and the commutant of $R_e$ is $G$-simple, then $R$ is a simple
ring. These results apply to $G$-crossed products in particular. A skew group
ring $R_e \rtimes_{\sigma} G$, where $R_e$ is commutative, is shown to be a
simple ring if and only if $R_e$ is $G$-simple and maximal commutative in $R_e
\rtimes_{\sigma} G$. As an interesting example we consider the skew group
algebra $C(X) \rtimes_{\tilde{h}} \mathbb{Z}$ associated to a topological
dynamical system $(X,h)$. We obtain necessary and sufficient conditions for
simplicity of $C(X) \rtimes_{\tilde{h}} \mathbb{Z}$ with respect to the
dynamics of the dynamical system $(X,h)$, but also with respect to algebraic
properties of $C(X) \rtimes_{\tilde{h}} \mathbb{Z}$. Furthermore, we show that
for any strongly $G$-graded ring $R$ each nonzero ideal of $R$ has a nonzero
intersection with the commutant of the center of the neutral component.
