  We introduce a new family of superalgebras which should be considered as a
super version of the Khovanov-Lauda-Rouquier algebras. Let $I$ be the set of
vertices of a Dynkin diagram with parity. To this data, we associate a family
of graded superalgebras, the quiver Hecke superalgebras. When there are no odd
vertices, these algebras are nothing but the usual Khovanov-Lauda-Rouquier
algebras. We then define another family of graded superalgebras, the quiver
Hecke-Clifford superalgebras, and show that they are weakly Morita
superequivalent to the quiver Hecke superalgebras. Moreover, we prove that the
affine Hecke-Clifford superalgebras, as well as their degenerate version, the
affine Sergeev superalgebras, are isomorphic to the quiver Hecke-Clifford
superalgebras after a completion.
