  Infinite-dimensional Galilean conformal algebras can be constructed by
contracting pairs of symmetry algebras in conformal field theory, such as
$W$-algebras. Known examples include contractions of pairs of the Virasoro
algebra, its $N=1$ superconformal extension, or the $W_3$ algebra. Here, we
introduce a contraction prescription of the corresponding operator-product
algebras, or equivalently, a prescription for contracting tensor products of
vertex algebras. With this, we work out the Galilean conformal algebras arising
from contractions of $N=2$ and $N=4$ superconformal algebras as well as of the
$W$-algebras $W(2,4)$, $W(2,6)$, $W_4$, and $W_5$. The latter results provide
evidence for the existence of a whole new class of $W$-algebras which we call
Galilean $W$-algebras. We also apply the contraction prescription to affine Lie
algebras and find that the ensuing Galilean affine algebras admit a Sugawara
construction. The corresponding central charge is level-independent and given
by twice the dimension of the underlying finite-dimensional Lie algebra.
Finally, applications of our results to the characterisation of structure
constants in $W$-algebras are proposed.
