In this thesis I have analysed several mathematical models, which represent the dynamics of genetic regulatory networks. Methods of bifurcation analysis and direct numerical simulations were employed to study the biological phenomena that can occur due to the presence of time delays, such as stable periodic oscillations induced by Hopf bifurcations. To highlight the biological implications of time-delayed systems, different models of genetic regulatory networks as relevant to the onset and development of cancer were studied in detail, as well as genetic regulatory networks which describe the effects of transcription factors in the immune system. A network of an oscillator coupled with a switch was explored, as systems such as these are prevalent in genetic regulatory networks. The effects of time delays on its oscillatory and bistable behaviour were then investigated, the results of which were compared with available results from the literature.