Advances in Geographical Information Systems (GIS) have led to enormous recent growth in spatiotemporal databases and associated statistical modeling with applications in various scientific disciplines, including environmental monitoring, ecological systems, forestry, hydrology, meteorology and public health. After inferring on a spatiotemporal process for a given dataset, inferential interest may turn to estimating rates of change, or gradients, over space and time. The primary focus of this thesis is to further develop the methodology required for statistical inference on areally-referenced temporal and spatiotemporal gradient processes. We begin by first departing from the rather rich literature in space-time modeling by considering the setting where space is discrete but time is continuous. Our major objective here is to carry out inference on gradients of a temporal process in our dataset of monthly county level asthma hospitalization rates in the state of California, while also accounting for spatial similarities of the temporal process across neighboring counties. In addition to using a more flexible stochastic process embedded within a dynamic Markov random field framework that permits inference on the temporal gradient process, we also develop methods for allowing region-specific variance components, leading to variable smoothing in our spatial regions. We then move to the continuous space, continuous time setting. Here, we develop, within a flexible spatiotemporal process model setting, a framework to estimate arbitrary directional gradients over space at any given timepoint, temporal derivatives at any given spatial location and, finally, mixed spatiotemporal gradients that reflect rapid change in spatial gradients over time and vice-versa. After illustrating the use of our methodology on a dataset comprising daily PM2.5 concentrations in California, we show how the method can be implemented to analyze highly censored data (e.g., data below detectable limits) and apply these methods to data collected during the cleanup efforts of the Deepwater Horizon (BP) oil spill. Through the use of these methods, we believe researchers can gain significant insight into potentially important spatiotemporally varying risk factors that may as of yet be unknown (or at least not accounted for). Furthermore, the gradient process in and of itself can provide valuable information, for instance by being adapted to alert public health officials of dramatically rising pollution levels in a particular region, potentially leading to a reduction in exposure and, ultimately, a reduction in the incidence of poor health outcomes.