This paper proposes a methodology for solving the curved path following problem for underactuated vehicles under unknown ocean current influence using deep reinforcement learning. Three dynamic models of high complexity are employed to simulate the motions of a mariner vessel, a container vessel and a tanker. The policy search algorithm is tasked to find suitable steering policies, without any prior info about the vessels or their environment. First, we train the algorithm to find a policy for tackling the straight line following problem for each of the simulated vessels and then perform transfer learning to extend the policies to the curved-path case. This turns out to be a much faster process compared to training directly for curved paths, while achieving indistinguishable performance. Index Terms-Deep reinforcement learning, path following, transfer learning, marine control systems, unknown disturbances