  In this thesis we discuss non-perturbative phenomena emerging in gauge and in
string/supergravity theories. We compute the partition function of 5D minimal
supersymmetric U(1) gauge theory with extra adjoint matter in general
Omega-background. These N=2^* theories are massive deformations of N=4 SYM and
can be thought of as a minimal supersymmetric five dimensional theory
compactified on a circle. Such partition functions encode very rich topological
information. We show that this partition function at some special values of the
parameters directly reproduces the generating function for the Poincare`
polynomial of the moduli space of instantons. We discuss the basic aspects of
worldsheet and penta-brane instantons as well as (unoriented) D-brane
instantons and threshold corrections to BPS-saturated couplings in superstring
theories. We consider non-perturbative superpotentials generated by `gauge' and
`exotic' instantons living on D3-branes at orientifold singularities. We also
discuss the interplay between worldsheet and D-string instantons on T^4/Z_2. We
focus on a 4-fermi amplitude and offer a multi D-string instanton
interpretation. We discuss also AdS/CFT correspondence and the role of
instantons there. We concentrate on AdS4/CFT3 correspondence. We revisit
Kaluza-Klein compactification of 11-d supergravity on S^7/Z_k using group
theory techniques that may find application in other flux vacua with internal
coset spaces. Among the SO(2) neutral states, we identify marginal deformations
and fields that couple to the recently discussed worldsheet instanton of Type
IIA on CP^3. We also discuss charged states, dual to monopole operators, and
the Z_k projection of the Osp(8|4) singleton and its tensor products. We show
that the doubleton spectrum may account for N =6 higher spin symmetry
enhancement in the limit of vanishing 't Hooft coupling in the boundary CS
theory.
