Dynamical systems, expressed as differential equations, serve as fundamental tools across various scientific disciplines for describing the temporal or spatial dynamics of real-world processes. Despite the increasing abundance of data from such processes, deriving the underlying physical laws analytically has grown more challenging. Consequently, statisticians and applied mathematicians have turned to developing techniques to estimate the differential equations governing temporally or spatially evolving systems from time series data. This dissertation contributes to this field by presenting novel advancements in both parameter estimation for differential equations of a known functional form and equation discovery for systems of an unknown form. The first method introduced is a Bayesian regression algorithm tailored for fitting a specific system of differential equations to time-course gene expression data. Subsequently, the resulting model is utilized to cluster genes based on the similarity of their temporal behavior. Following this, a regularized regression-based approach is presented for recovering systems of ordinary differential equations from time series data. Notably, this method offers classical uncertainty estimates in the form of confidence intervals and hypothesis tests for each term in the reconstructed equation. This addresses a pertinent issue in the literature on data-driven dynamical system recovery, namely how to quantify uncertainty in a computationally efficient and interpretable manner. Finally, an extension of this methodology to partial differential equations recovered from spatiotemporal data is proposed and contextualized within the broader field of scientific machine learning. Detailed discussions on future extensions of the proposed methods are provided, and software packages implementing these methods are made publicly accessible.