Short-term forecasting of ocean surface currents improves our understanding of ocean dynamics and has numerous applications such as planning ship routes over multiple days. Satellites measure various physical properties of the ocean in real-time. Observations such as sea surface temperature (SST), chlorophyll concentration, and sea surface height (SSH) provide indirect measurements of ocean surface currents. Direct measurement of ocean currents is possible using in-situsensors, but these measurements are sparse and noisy. Due to lack of sufficient ground truth data, neural methods estimating ocean currents are often trained on synthetic data produced from physically realistic ocean simulations derived from fluid dynamics equations, with associated simulated satellite observations. Real satellite data can then be used to fine-tune or evaluate the model.We use data from numerical simulations in conjunction with real-world satellite observations and in-situ measurements to train and validate a deep-learning model to forecast ocean currents on several days. Our model builds upon previous work byfocusing on forecasting ocean surface current fields in addition to only SSH fields. Furthermore, this approach is observation-driven rather than an explicit physical forecast. In a first stage, we train a convolutional neural network to forecast currentsusing simulated satellite observations and ground truth data from a numerical model. In a second transfer-learning stage, we use real altimetric observations and in-situ measurements to fine-tune our model. We evaluate our ocean current forecast using both numerical simulations and real in-situ measurements from drifters and on-board instruments from ships.