  In this paper we describe explicit $L_\infty$ algebras modeling the rational
homotopy type of any component of the spaces $\map(X,Y)$ and $\map^*(X,Y)$ of
free and pointed maps between the finite nilpotent CW-complex $X$ and the
finite type nilpotent CW-complex $Y$. When $X$ is of finite type, non
necessarily finite, we also show that the algebraic covers of these $L_\infty$
algebras model the universal covers of the corresponding mapping spaces.
