This dissertation analyzes the dynamics of the individual components in two specific biological networks in order to understand how these components interact to produce observed cellular behavior. First, we use an integrated experimental-computational approach to analyze the dynamical response of a synthetic positive feedback network in individual mammalian cells. Using flow cytometry, we observe a switch-like activation of the network with variable delay times in individual cells. In agreement with a stochastic model of the network, we find that increasing the strength of the positive feedback results in a decrease in the mean delay time and a more coherent activation of individual cells. The results of this work are important for gaining insight into biological processes such as cell cycle regulation and apoptosis which rely on positive feedback to generate switch-like responses and may also facilitate the development of engineered mammalian control systems. Second, we use computational modeling to study the dynamics of the NF-kappaB signaling pathway that governs important cellular processes such as inflammation and the immune response. Because the NF-kappaB pathway contains over 100 reactions, the complexity of this signaling network is enormous. Here, we utilize a modeling approach which replaces the complicated cascades of individual biochemical reactions by few compound but delayed reactions. We utilize both deterministic and stochastic formulations of our model to interrogate the negative feedback loops that regulate the dynamic activity of NF- kappaB. In agreement with our experiments, we find that the response of the dual-feedback circuit is tuned to minimize oscillations. Further, we reveal two important features of the dual-feedback-loop architecture that may explain its evolutionary advantage over no or single- feedback systems: first, it ensures a highly sensitive initial response while allowing for temporally graded outputs; and second, it minimizes stochastic fluctuations and leads to a robust response to incoming signals. In conclusion, this dissertation investigates the behavior of both an artificial gene regulatory network and a naturally -occurring signaling network. This work involves the utilization of both computational and experimental techniques to gain insight into the dynamics of regulatory networks in mammalian systems.