We introduce a novel approach to automatic unstructured mesh generation using machine learning to predict an optimal Finite element mesh for a previously unseen problem. The framework that we have developed is based around training an artificial neural network (ANN) to guide standard mesh generation software, based upon a prediction of the required local mesh density throughout the domain. We describe the training regime that is proposed, based upon the use of a posteriori error estimation, and discuss the topologies of the ANNs that we have considered. We then illustrate performance using two standard test problems, a single elliptic partial dierential equation (PDE) and a system of PDEs associated with linear elasticity. We demonstrate the eective generation of high quality meshes for arbitrary polygonal geometries and a range of material parameters, using a variety of user-selected error norms.