We introduce the {\it Ellis semigroup} of a nonautonomous discrete dynamical
system (X,f1,∞) when X is a metric compact space. The underlying
set of this semigroup is the pointwise closure of \{f\sp{n}_1 \, |\, n\in
\mathbb{N}\} in the space X\sp{X}.
By using the convergence of a sequence of points with respect to an
ultrafilter it is possible to give a precise description of the semigroup and
its operation. This notion extends the classical Ellis semigroup of a discrete
dynamical system. We show several properties that connect this semigroup and
the topological properties of the nonautonomous discrete dynamical system