In recent decades there has been tremendous growth in new statistical methods and applications for modeling random events occurring in space and time. There is a positive outlook on the demand for research in this field for the coming decades as space- and time-referenced event data will become more commonly available and high in size and resolution.

This dissertation makes methodological contributions in non-parametric and semi-parametric self-exciting point processes models and their application to infectious disease spread and quantitative criminology. For infectious diseases we demonstrate that point process models can be an effective tool for real-time descriptions and forecasts of an outbreak by comparing its performance to traditional compartmental models. We introduce a purely infection-driven, non-stationary point process model and its estimation. We propose a non-parametric implementation of the Recursive Hawkes model. In dealing with gang crime event data, we address the long standing challenge of distinguishing spatial and temporal inhomogeneity with true triggering, as well as evaluating event-based treatments that are non-randomized due to practical and ethical considerations. To this end, we propose a new method to non-parametrically incorporate spatial covariates in the background rate of crimes. We also introduce a sub-sampling procedure to evaluate non-randomized, clustered treatments in order to generate synthetic controls to improve causal interpretation. We assess this procedure with simulation studies