  We define and study supercharacters of the classical finite unipotent groups
of symplectic and orthogonal types (over any finite field of odd
characteristic). We show how supercharacters for groups of those types can be
obtained by restricting the supercharacter theory of the finite unitriangular
group, and prove that supercharacters are orthogonal and provide a partition of
the set of all irreducible characters. We also describe all irreducible
characters of maximum degree in terms of the root system, and show how they can
be obtained as constituents of particular supercharacters.
