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The family of 2-level matroids, that is, matroids whose base polytope is2-level, has been recently studied and characterized by means of combinatorialproperties. 2-level matroids generalize series-parallel graphs, which have beenalready successfully analyzed from the enumerative perspective.  We bring to light some structural properties of 2-level matroids and exploitthem for enumerative purposes. Moreover, the counting results are used to showthat the number of combinatorially non-equivalent (n-1)-dimensional 2-levelpolytopes is bounded from below by $c \cdot n^{-5/2} \cdot \rho^{-n}$, where$c\approx 0.03791727 $ and $\rho^{-1} \approx 4.88052854$.

Many 2-level polytopes from matroids