  We show how to construct measures on Banach manifolds associated to
supersymmetric quantum field theories. These measures are mathematically
well-defined objects inspired by the formal path integrals appearing in the
physics literature on quantum field theory. We give three concrete examples of
our construction. The first example is a family $\mu_P^{s,t}$ of measures on a
space of functions on the two-torus, parametrized by a polynomial $P$ (the
Wess-Zumino-Landau-Ginzburg model). The second is a family $\mu_\cG^{s,t}$ of
measures on a space $\cG$ of maps from $\P^1$ to a Lie group (the
Wess-Zumino-Novikov-Witten model). Finally we study a family $\mu_{M,G}^{s,t}$
of measures on the product of a space of connection s on the trivial principal
bundle with structure group $G$ on a three-dimensional manifold $M$ with a
space of $\fg$-valued three-forms on $M.$
  We show that these measures are positive, and that the measures
$\mu_\cG^{s,t}$ are Borel probability measures. As an application we show that
formulas arising from expectations in the measures $\mu_\cG^{s,1}$ reproduce
formulas discovered by Frenkel and Zhu in the theory of vertex operator
algebras. We conjecture that a similar computation for the measures
$\mu_{M,SU(2)}^{s,t},$ where $M$ is a homology three-sphere, will yield the
Casson invariant of $M.$
