Gaussian graphical models are widely popular for studying the conditional dependence among random variables. By encoding conditional dependence as an undirected graph, these models provide interpretable representations and insightful visualizations of the relationships among variables. However, time series data often violate the assumptions of Gaussian graphical models. In time series, the data are often not iid; the graphs can evolve over time, with changes occurring at unknown time points. We first extend Bayesian graphical models to time series data with heavy tailed characteristics. We introduce a Dynamic and Robust Gaussian Graphical model, which is able to identify dynamics in the graph, share information across time, and estimate graphs from highly contaminated data. We then consider the scenario where the data are less contaminated and close to smooth curves. We introduce a Dynamic Bayesian Functional Graphical Model, where the observed data is viewed as realizations of random functions varying over a continuum of time. Unlike the dynamic and robust time series model, each node in the functional graphical model represents a function. The model inserts a change point in time and estimates two different graphs before and after the change point. The proposed methods demonstrate excellent graph estimation for simulated data with improvements over existing graphical models. We apply these methods in various applications, including gesture tracing data, futures return data and sea surface temperature data, and discover meaningful edges and dynamics.