The Lyubeznik numbers are a family of invariants of a local ring containing a field that detect many ring-theoretic properties, and also have topological and geometric interpretations.  Motivated by their utility, in joint work with Luis Núñez-Betancourt, we define a new family of invariants of any local ring with characteristic p>0 residue field; in particular, they are defined for local rings of mixed characteristic.  For local rings of equal characteristic p>0, both the Lyubeznik numbers and these new invariants are defined.  In some cases, they agree, but we give an example where they differ.