This paper presents a new approach for optimising the design of tidal stream turbine farms. In this approach, the turbine farm is represented by a turbine density function that specifies the number of turbines per unit area and an associated continuous locally-enhanced bottom friction field. The farm design question is formulated as a mathematical optimisation problem constrained by the shallow water equations and solved with efficient, gradient-based optimisation methods. The resulting method is accurate, computationally efficient, allows complex installation constraints, and supports different goal quantities such as to maximise power or profit. The outputs of the optimisation are the optimal number of turbines, their location within the farm, the overall farm profit, the farm's power extraction, and the installation cost. We demonstrate the capabilities of the method on a validated numerical model of the Pentland Firth, Scotland. We optimise the design of four tidal farms simultaneously, as well as individually, and study how farms in close proximity may impact upon one another. Crown Copyright (C) 2016 Published by Elsevier Ltd. All rights reserved.