  In 1979, Nishizeki and Baybars showed that every planar graph with minimum
degree 3 has a matching of size $\frac{n}{3}+c$ (where the constant $c$ depends
on the connectivity), and even better bounds hold for planar graphs with
minimum degree 4 and 5. In this paper, we investigate similar matching-bounds
for {\em 1-planar} graphs, i.e., graphs that can be drawn such that every edge
has at most one crossing. We show that every 1-planar graph with minimum degree
3 has a matching of size at least $\frac{1}{7}n+\frac{12}{7}$, and this is
tight for some graphs. We provide similar bounds for 1-planar graphs with
minimum degree 4 and 5, while the case of minimum degree 6 and 7 remains open.
