  A superspecial curve is a (non-singular) curve over a field of positive
characteristic whose Jacobian variety is isomorphic to a product of
supersingular elliptic curves over the algebraic closure. It is known that for
given genus and characteristic, there exist only finitely many superspecial
curves, up to isomorphism over an algebraically closed field. In this article,
we give a brief survey on results of counting isomorphism classes of
superspecial curves. In particular, this article summarizes some recent results
in the case of genera four and five, obtained by the author and S.\ Harashita.
We also survey results obtained in a joint work with Harashita and E.\ W.\
Howe, on the enumeration of superspecial curves in a certain class of
non-hyperelliptic curves of genus four.
