DescriptionIn this thesis we develop and apply novel techniques for analyzing BPS spectra of supersymmetric quantum field theories of class S. By a combination of wall-crossing, spectral networks and quiver methods we explore the BPS spectra of higher rank four-dimensional N = 2 super Yang-Mills, uncovering surprising new phenomena. Focusing on the SU(3) case, we prove the existence of wild BPS spectra in field theory, featuring BPS states of higher spin whose degeneracies grow exponentially with the energy. The occurrence of wild BPS states is surprising because it appears to be in tension with physical expectations on the behavior of the entropy as a function of the energy scale. The solution to this puzzle comes from realizing that the size of wild BPS states grows rapidly with their mass, and carefully analyzing the volume-dependence of the entropy of BPS states. We also find some interesting structures underlying wild BPS spectra, such as a Regge-like relation between the maximal spin of a BPS multiplet and the square of its mass, and the existence of a universal asymptotic distribution of spin-j irreps within a multiplet of given charge. We also extend the spectral networks construction by introducing a refinement in the topological classification of 2d-4d BPS states, and identifying their spin with a topological invariant known as the “writhe of soliton paths”. A careful analysis of the 2d-4d wall-crossing behavior of this refined data reveals that it is described by motivic Kontsevich-Soibelman transformations, controlled by the Protected Spin Character, a protected deformation of the BPS index encoding the spin of BPS states. Our construction opens the way for the systematic study of refined BPS spectra in class S theories. We apply it to several examples, including ones featuring wild BPS spectra, where we find an interesting relation between spectral networks and certain functional equations. For class S theories of A1 type, we derive an alternative technique for computing generating functions of 2d-4d BPS spectra, based on the topological data of an ideal triangulation of the Riemann surface defining the theory. We provide a set of building blocks and corresponding rules, from which the 2d-4d spectra of a vast class of theories can be algorithmically recovered. Finally, we present previously unpublished exact results on the BPS spectrum of the SU(2) N = 2∗ theory, and briefly comment on its wall crossing.