Topological superconductors are fully gapped in the bulk but host Majorana modes on their
boundaries. We extend this notion to a new class of superconductors, second-order topological
superconductors, that host Majorana modes instead on second-order boundaries, i.e., corners of a
two-dimensional system and hinges for a three-dimensional system.
Here we propose two general scenarios in which second-order topological superconductivity
can be realized. First, we show that $p_x+ip_y$-wave pairing in a (doped) Dirac semimetal
in two dimensions with four mirror symmetric Dirac nodes realizes second-order topological
superconductivity. Second, we show that the four Dirac nodes can also come from the BdG spectrum
of a $d$-wave superconductor. In this scenario, with an additional $p$-wave pairing that gaps out the
Dirac nodes, the system realizes second-order topological superconductivity as well. We show that
these exotic superconducting states can be intrinsically realized in a metallic system with electronic
interactions, or induced by proximity effect.