The movement of oceanic water has important consequences for a variety of applications, such as climate change, biological productivity, sea-level change, weather forecasting, and many others. This book addresses the problem of inferring the state of the ocean circulation, understanding it dynamically, and even forecasting it through a quantitative combination of theory and observation. It focuses on so-called inverse methods and related methods of statistical inference. Both time-independent and time-dependent problems are considered, including Gauss-Markov estimation, sequential estimators and adjoint/Pontryagin principle methods. This book is intended for use as a graduate level text for students of oceanography and other related fields. It will also be of interest to working physical oceanographers.