  The standard procedure for quantizing gauge fields is the Faddeev-Popov
quantization, which performs gauge fixing in the path integral formulation and
introduces additional ghost fields. This approach provides the foundation for
calculations in quantum Yang-Mills theory. However, in 1978, Vladimir Gribov
showed that the gauge-fixing procedure was incomplete, with residual gauge
copies (called Gribov copies) still entering the path integral even after gauge
fixing. These copies impact the infrared behavior of the theory and modify
gauge-dependent quantities, such as gluon and ghost propagators, as they
represent redundant integrations over gauge-equivalent configurations.
Furthermore, their existence breaks down the Faddeev-Popov prescription at a
fundamental level. To partially resolve this, Gribov proposed restricting the
path integral to the Gribov region, which alters the gluon propagator
semiclassically in a way that points to gluon confinement in the Yang-Mills
theory.
  In this thesis, we comprehensively study the Gribov problem analytically.
After reviewing Faddeev-Popov quantization, the BRST symmetry of the complete
Lagrangian and the Gribov problem in depth, we detail Gribov's semi-classical
resolution involving restriction of the path integral to the Gribov region,
outlining its effects on the theory. Further, we elucidate stochastic
quantization prescription for quantizing the gauge fields. This alternate
quantization prescription hints towards a formalism devoid of the Gribov
problem, making it an interesting candidate for quantizing and studying the
non-perturbative regime of gauge theories.
