  We describe the structure of the symbolic powers $I^{(\ell)}$ of the
Stanley-Reisner ideals, and cover ideals, $I$, of matroids. We (a) prove a
structure theorem describing a minimal generating set for every $I^{(\ell)}$;
(b) describe the (non--standard graded) symbolic Rees algebra
$\mathcal{R}_s(I)$ of $I$ and show its minimal algebra generators have degree
at most ht $I$; (c) provide an explicit, simple formula to compute the largest
degree of a minimal algebra generator of $\mathcal{R}_s(I)$; (d) provide
algebraic applications, including formulas for the symbolic defects of $I$, the
initial degree of $I^{(\ell)}$, and the Waldschmidt constant of $I$; (e)
provide a new algorithm allowing fast computations of very large symbolic
powers of $I$.
  One of the by-products is a new characterization of matroids in terms of
minimal generators of $I^{(\ell)}$ for some $\ell\geq 2$. In particular, it
yields a new, simple characterization of matroids in terms of the minimal
generators of $I^{(2)}$. This is the first characterization of matroids in
terms of $I^{(2)}$, and it complements a celebrated theorem by Minh-Trung,
Varbaro, and Terai-Trung which requires the investigation of homological
properties of $I^{(\ell)}$ for some $\ell\geq 3$.
