In this thesis we investigate properties of equilibrium and non-equilibrium systems by means of renormalization
group (RG) analysis. In the study of the d-dimensional Coulomb gas we have formulated
a continuum model from the underlying hyper-cubic lattice and employed the irreducible differential
formulation of the Wilson RG.We have identified a Thouless-Kosterletz transition in d=2 and found
no non-trivial fixed points for d>2. As an example of a non-equilibrium system, we have investigated
properties of quasi-neutral plasmas which are governed by stochastic magnetohydrodynamic
(MHD) equations. The present method is based upon the Martin-Siggia-Rose field-theory formulation
of stochastic dynamics. We develop a diagrammatic representation for the theory and carry out
a momentum-shell RG of Wilson-Kadanoff type. An infinite set of diagrams is identified which are
marginal in the RG sense. We have shown, in accordance with previous literature, that the same problem
arises for the randomly-forced Navier-Stokes equation. The problem of marginal variables can
be suppressed by working near equilibrium, where stochastic forcing represents thermal fluctuations.
In a similar manner we have considered regimes when MHD equations are subject either to kinetic or
magnetic forcing only. In such models the macroscopic limit can be taken such that all marginal terms
are irrelevant and the dynamics is governed by linear equations. Furthermore, non-trivial fixed points
are identified in such regimes and limiting values of either kinematic viscosity or magnetic diffusivity
are derived. A consistent description of MHD dynamics far from equilibrium is still absent. We highlight
some of the aspects of the functional integral formulation with regards to the symmetries of the
system and propose possible ways in which the system can be studied non-pertubatively.