This is the Ph.D. dissertation of the author. The project has been motivated by the conjecture that the Hopkins-Miller tmf spectrum can be described in terms of `spaces' of conformal field theories. In this dissertation, spaces of field theories are constructed as classifying spaces of categories whose objects are certain types of field theories. If such a category has a symmetric monoidal structure and its components form a group, by work of Segal, its classifying space is an infinite loop space and defines a cohomology theory. This has been carried out for two classes of field theories: (i) For each integer n, there is a category SEFT_n whose objects are the Stolz-Teichner (1|1)-dimensional super Euclidean field theories of degree n. It is proved that the classifying space |SEFT_n| represents degree-n K or KO cohomology, depending on the coefficients of the field theories. (ii) For each integer n, there is a category AFT_n whose objects are a kind of (2|1)-dimensional field theories called `annular field theories,' defined using supergeometric versions of circles and annuli only. It is proved that the classifying space |AFT_n| represents the degree-n elliptic cohomology associated with the Tate curve. To the author's knowledge, this is the first time the definitions of low-dimensional supersymmetric field theories are given in full detail.