Non-perturbative aspects of supersymmetric gauge theories and string theory

In this dissertation we explore various exact non-perturbative results in supersymmetric gauge theories. We also discuss some issues in strongly-coupled string theory such as the interactions of branes in a matrix model of M-theory and the use of D-branes as probes and as microscopic descriptions of black holes.; We begin with an investigation of {dollar}N=1{dollar} supersymmetric theories based on exceptional groups. The flat directions are most easily parameterized using their correspondence with gauge invariant polynomials. Symmetries and holomorphy tightly constrain the superpotentials, but due to multiple gauge invariants other techniques are needed for their full determination. We give an explicit treatment of {dollar}Gsb2{dollar} and find gaugino condensation for {dollar}Nsb{lcub}f{rcub}le 2,{dollar} and an instanton generated superpotential for {dollar}Nsb{lcub}f{rcub}=3.{dollar} The analogy with {dollar}SU(Nsb{lcub}c{rcub}{dollar}) gauge theories continues with modified and unmodified quantum moduli spaces for {dollar}Nsb{lcub}f{rcub}=4{dollar} and {dollar}Nsb{lcub}f{rcub}=5{dollar} respectively, and a non-Abelian Coulomb phase for {dollar}Nsb{lcub}f{rcub}ge6.{dollar} Electric variables suffice to describe this phase over the full range of {dollar}Nsb{lcub}f{rcub}.{dollar}; Next we apply the method of confining phase superpotentials to derive the exact quantum moduli space of {dollar}N=2{dollar} supersymmetric Yang-Mills theory with the exceptional gauge group {dollar}Gsb2.{dollar} Our findings are consistent with the spectral curve of the periodic Toda lattice, but do not agree with the hyperelliptic curve suggested previously in the literature. We also apply the method to theories with fundamental matter, treating both the example of SO(5) and {dollar}Gsb2.{dollar}; Turning to string theory, we consider eight-brane configurations in M(atrix) theory and compute their interaction potentials with gravitons, membranes, and four-branes. We compare these results with the interactions of D8-branes with D0-branes, D2-branes, and D4-branes in IIA string theory. We find agreement between the two approaches for eight-brane interactions with two-branes and four-branes. A discrepancy is noted in the case with zero-branes.; Finally, we consider configurations of D6-branes with D0-brane charge and compute interaction potentials with various D-brane probes using a 1-loop open string calculation. These results are compared to a supergravity calculation using the solution of an extremal black hole carrying 0-brane and 6-brane charge. We find precise agreement of the long distance interaction potentials and explore the short distance behavior of the D-brane configuration.