This work investigates fundamental two-dimensional vortex pair dynamics in unstratified and stably stratified environments through numerical and analytical techniques. The study focuses on two main topics: (i) vortex interaction and merging of co-rotating vortex pairs and (ii) internal wave generation by co-rotating and counter-rotating vortex pairs. Two-dimensional vortex merging in a viscous fluid is studied using numerical simulations. Analysis of the ideal case of two equal co-rotating vortices (symmetric pair) identifies the basic underlying physics of vortex merger. Through the interaction of the vorticity gradient and the mutually induced strain rate near the central hyperbolic point, a tilt in vorticity contours is established. This leads to core detrainment and the entrainment of core fluid into the exchange band, which transforms the flow into a single vortex. In the case of the asymmetric (unequal strength) vortex pair, the disparity in the deformation rates between the vortices alters the interaction. A critical value for a strain rate parameter characterizing the establishment of core detrainment is determined. The onset of merging is associated with the achievement of the critical strain by both vortices and a generalized merging criterion is formulated. A classification scheme of the various viscous vortex interactions is developed. Results for the symmetric, horizontally oriented vortex pair in a weakly stratified fluid provide further insight on vortex merging. The effects of weak stratification depend on the ratio of the diffusive time scale to the turnover time, i.e., the Reynolds number. A crossover Reynolds number is found, above which convective merging is accelerated with respect to unstratified flow and below which it is delayed. The generation of internal waves by {\it horizontally} orientated co-rotating and counter-rotating vortex pairs is studied. Linearized inviscid equations are derived that describe the internal wave, vorticity and energy fields. These solutions are compared with nonlinear numerical viscous simulations in moderately and strongly stratified environments. Through evaluation of the energy field, the time at which the flow reaches a steady state for strongly stratified flows is found, along with a characterization of the regimes of strongly and moderately stratified environments.