  In this paper we introduce a product-like operation that generalizes the
construction of generalized Sierpi\'nski graphs. Let $G,H$ be graphs and let
$f: V(G) \to V(H)$ be a function. Then the Sierpi\'nski product of $G$ and $H$
with respect to $f$ is defined as a pair $(K,\varphi)$, where $K$ is a graph on
the vertex set $V(G) \times V(H)$ with two types of edges:
  -- $\{(g,h),(g,h')\}$ is an edge in $K$ for every $g\in V(G)$ and every
$\{h,h'\}\in E(H)$,
  -- $\{(g,f(g'),(g',f(g))\}$ is an edge in $K$ for every edge $\{g,g'\} \in
E(G)$; and $\varphi: V(G) \to V(K)$ is a function that maps every vertex $g \in
V(G)$ to the vertex $(g,f(g)) \in V(K)$. Graph $K$ will be denoted by
$G\otimes_f H$. Function $\varphi$ is needed to define the product of more than
two factors. By applying this operation $n$ times to the same graph we obtain
the $n$-th generalized Sierpi\'nski graph.
  Some basic properties of the Sierpi\'nski product are presented. In
particular, we show that $G \otimes_f H$ is connected if and only if both $G$
and $H$ are connected and we present some necessary and sufficient conditions
that $G,H$ must fulfill in order for $G \otimes_f H$ to be planar. As for
symmetry properties, we show which automorphisms of $G$ and $H$ extend to
automorphisms of $G \otimes_f H$. In many cases we can also describe the whole
automorphism group of $G\otimes_f H$.
