  Symplectic $Q$-functions are a symplectic analogue of Schur $Q$-functions and
defined as the $t=-1$ specialization of Hall--Littlewood functions associated
with the root system of type $C$. In this paper we prove that symplectic
$Q$-functions share many of the properties of Schur $Q$-functions, such as a
tableau description and a Pieri-type rule. And we present some positivity
conjectures, including the positivity conjecture of structure constants for
symplectic $P$-functions. We conclude by giving a tableau description of
factorial symplectic $Q$-functions.
