Topics in the superconformal and defect conformal bootstrap

Conformal field theory has varied applications, ranging from critical phenomena in statistical and condensed matter physics, to formal aspects of quantum gravity. In recent years, the conformal bootstrap has become one of the most promising frameworks to study conformal field theory, both at the perturbative and non-perturbative level. In this thesis, we consider two extensions of the conformal bootstrap: the addition of supersymmetry, and the addition of conformal defects. Both of these extensions have rich physical applications, that we discuss in the two main parts of the thesis. In the first part, we present a pedagogical introduction to conformal field theory, the bootstrap program, supersymmetry and conformal defects. In the second part, we discuss several applications of these general methods.