We study, by using the theory of algebraic $\cD$-modules, the local cohomology modules supported on a monomial ideal $I$ of the polynomial ring $R=k[x_1,\dots,x_n]$, where $k$ is a field of characteristic zero. We compute the characteristic cycle of $H_I^r(R)$ and $H_{{\p}}^p(H_I^r(R))$, where ${\p}$ is an homogeneous prime ideal of $R$. By using these results we can describe the support of these modules, in particular we can decide when the local cohomology module $H_I^r(R)$ vanishes in terms of the minimal primary decomposition of the monomial ideal $I$, compute the Bass numbers of $H_I^r(R)$ and describe its associated primes. The characteristic cycles also give some invariants of the ring $R/I$. We use these invariants to compute the Hilbert function of $R/I$, the minimal free resolutions of squarefree monomial ideals and the cohomology groups of the complement of an arrangement of linear varieties given by the monomial ideal $I$. Finally, we determine the local cohomology modules by using the category introduced by Galligo-Granger-Maisonobe \cite{GGM85b} and compute its Hilbert function.