  We address the problem of identifying the (nonstationary) quantum systems
that admit supersymmetric dynamical invariants. In particular, we give a
general expression for the bosonic and fermionic partner Hamiltonians. Due to
the supersymmetric nature of the dynamical invariant the solutions of the
time-dependent Schr\"odinger equation for the partner Hamiltonians can be
easily mapped to one another. We use this observation to obtain a class of
exactly solvable time-dependent Schr\"odinger equations. As applications of our
method, we construct classes of exactly solvable time-dependent generalized
harmonic oscillators and spin Hamiltonians.
