  We introduce the notion of a crossed product of an algebra by a coalgebra
$C$, which generalises the notion of a crossed product by a bialgebra
well-studied in the theory of Hopf algebras. The result of such a crossed
product is an algebra which is also a right $C$-comodule. We find the necessary
and sufficient conditions for two coalgebra crossed products be equivalent. We
show that the two-dimensional quantum Euclidean group is a coalgebra crossed
product. The paper is completed with an appendix describing the dualisation of
construction of coalgebra crossed products.
