  We demonstrate that the spectrum of any consistent string theory in $D$
dimensions must satisfy a number of supertrace constraints: $ Str~M^{2n}=0 $
for $0 \leq n < D/2-1$, integer $n$. These results hold for a large class of
string theories, including critical heterotic strings. For strings lacking
spacetime supersymmetry, these supertrace constraints will be satisfied as a
consequence of a hidden ``misaligned supersymmetry'' in the string spectrum.
[Talk given by R.C.M. at Strings '95; to appear in Proceedings]
