In the search for a geometric model for elliptic cohomology and the Witten genus, Stolz and Teichner have defined a class of supersymmetric Euclidean field theories in various superdimensions d|delta. With collaborators, they have shown that 1|1-dimensional theories encode the A^-genus and KO-theory, and in dimension 2|1 they expect similar constructions to lead to the Witten genus and TMF, the universal elliptic cohomology theory. In this thesis, we investigate super Euclidean field theories in a variety of dimensions with the goal of understanding their role in algebraic topology. We focus on two aspects: (1) the appearance of invariants like the A^-genus and (2) the relationship between field theories and cohomology theories.

Beginning in the early 80s, physicists observed that partition functions of supersym- metric sigma models could frequently be identified with manifold invariants like the Euler characteristic, signature and A^-genus. Making these arguments precise culminated in the heat kernel proof of the index theorem. The first result in this thesis is a structural one: partition functions of supersymmetric sigma models always furnish manifold invariants. We prove this in Chapter 2. This leaves open the difficult question of how one might construct these supersymmetric sigma models.

Chapter 3 is a proof of the Chern-Gauss-Bonnet Theorem, which comes in two parts. We construct the 0|2-sigma model, which we know a priori leads to a manifold invariant. Then, via computations in supergeometry, we identify this invariant with the Euler characteristic. By coupling the sigma model to a constant times a Morse function we obtain a variety of expressions that compute the partition function, and by equating two of these we derive the Chern-Gauss-Bonnet formula.

In Chapter 4, we investigate the relationship between field theories and cohomology theories. Our results are various no-go theorems for when field theories give cocycles for cohomology theories for delta> 1. In dimensions 0|delta we are able to prove fairly strong statements to this effect, but for d > 0 we need to make assumptions motivated by our desired connection between invariants coming from sigma models and the putative cohomology theory coming from the space of field theories.