  In this paper we introduce Laguerre expansions to approximate vector-valued
functions expanding on the well-known scalar theorem. We apply this result to
approximate $C_0$-semi\-groups and resolvent operators in abstract Banach
spaces. We study certain Laguerre functions, its Laplace transforms and the
convergence of Laguerre series in Lebesgue spaces. The concluding section of
this paper is devote to consider some examples of $C_0$-semigroups: shift,
convolution and holomorphic semigroups where some of these results are
improved.
