  Non-perturbative formulations of field theories are essential to capture
intriguing physical phenomena, including confinement in QCD, spontaneous
supersymmetry breaking, and dynamical compactification in superstrings. Lattice
regularization provides a robust framework to study these non-perturbative
features through Euclidean path integrals. Conventionally, path integrals are
numerically evaluated using Monte Carlo methods, where the Boltzmann factor is
interpreted as a probability weight. However, complex actions in various
physical systems render the Boltzmann factor complex, leading to the sign
problem. The complex Langevin method overcomes the sign problem and can be used
to evaluate complex integrals. This thesis employs the complex Langevin method
to investigate various non-perturbative aspects of field-theoretic systems with
complex actions. We probe the possibility of spontaneous supersymmetry breaking
in the simplest realizations of supersymmetric field theories. These systems
generally have complex actions arising from a complex determinant of the
fermion operator. We studied various interesting classes of complex potentials,
including those exhibiting PT-symmetry. Another exciting aspect explored is the
dynamical compactification of extra dimensions in superstring theory. The IKKT
matrix model, in the large-N limit, is a conjectured formulation for the 10D
type IIB string theory. We employ the complex Langevin method to investigate
the Euclidean version of this matrix model, which has an inherent complex
Pfaffian, to probe the spontaneous breaking of SO(10) symmetry. The
investigations performed in this thesis suggest that the complex Langevin
method can successfully simulate non-perturbative aspects of quantum field
theories by taming the associated sign problem.
