Instances of vortex rings evolving in confined domains are observed in biological and engineering flows. The effect of non-axisymmetric confinement on the evolution of a vortex ring is studied in this thesis. The study is based on computer simulations employing the Lattice Boltzmann Method (LBM). Two flow domains are considered here: one is a fully closed wall enclosure and the other is periodic in streamwise direction. Vortex rings are initialized with a Gaussian distribution of vorticity within the core. The effect of relevant non-dimensional parameters involved in the problem, namely the confinement ratio, CR (representing relative size of the confinement), the aspect ratio of the vortex ring, AR (representing the slenderness of the initial vortex ring), and the Reynolds number, Re, are discussed in detail in the thesis. Due to its self-induced velocity, the original vortex ring moves inside the confinement. The evolution of the ring is characterized by several parameters such as kinetic energy, maximum core vorticity and circulation; the effect of wall interaction from confinement is also investigated. The presence of the confinement is observed to accelerate the decay of the vortex ring and the decay rate increases with an increase in CR. Furthermore, the symmetry of the ring breaks due to confinement and the initial ring shape is deformed. The asymmetry caused by the confinement is noted by measuring the vortex ring properties on two different planes, containing the axis of the ring, namely the Y-plane and the diagonal plane. The extent of deformation of the ring increases with both CR and Re. However, the patterns of deformation of the ring vary in different regimes of AR values. The introduction of small swirl velocity to the initial vortex ring is found to delay the asymmetry caused by the confinement.