A persistent carbene (also known as stable carbene) is an organic molecule whose natural resonance structure has a carbon atom with incomplete octet (a carbene), but does not exhibit the tremendous instability typically associated with such moieties.  The best-known examples and by far largest subgroup are the N-heterocyclic carbenes (NHC) (sometimes called Arduengo carbenes), in which nitrogen atoms flank the formal carbene.
Modern theoretical analysis suggests that the term "persistent carbene" is in fact a misnomer.  Persistent carbenes do not in fact have a carbene electronic structure in their ground state, but instead an ylide stabilized by aromatic resonance or steric shielding.  Excitation to a carbene structure then accounts for the carbene-like dimerization that some persistent carbenes undergo over the course of days.
Persistent carbenes in general, and Arduengo carbenes in particular, are popular ligands in organometallic chemistry.