Climate change is fundamentally ocean change. The ocean absorbs more than 90% of the Earth’s radiative imbalance and about 30% of anthropogenic CO_2 emissions, leading to ocean warming and acidification. Because of the formidable challenge of observing the full-depth global ocean circulation in its spatial detail and the many time scales of oceanic motions, numerical simulations play an essential role in quantifying patterns of climate variability and change. For the same reason, predictive capabilities are confounded by the high-dimensional space of uncertain inputs required to perform such simulations (initial conditions, model parameters and external forcings). Inverse methods optimally extract and blend information from observations and models. Parameter and state estimation, in particular, enable rigorously calibrated and initialized predictive models to optimally learn from sparse, heterogeneous data while satisfying fundamental equations of motion. A key enabling computational approach is the use of adjoint methods for solving a nonlinear least-squares optimization problem and the use of algorithmic differentiation for generating and maintaining derivative codes alongside a state-of-the-art ocean general circulation model. Emerging capabilities are the uncertainty propagation from the observations through the model to key oceanic metrics such as equator-to-pole oceanic mass and heat transport. Also of increasing interest is the application of optimal experimental design methods for developing effective observing systems. We argue that methods that are being developed in computational science and engineering at the interface of predictive data science, in particular those that are scalable to real-world problems, remain under-utilized in ocean climate modeling. Realizing their full potential involves considerable practical hurdles in the context of high-performance computing, but it is indispensable for advancing simulation-based contributions as we enter the UN Decade of Ocean Science for Sustainable Development.