Numerical solution of differential equations on the surface of the sphere requires grid generation. Examples include numerical simulations of mantle convection, weather, and climate. Because of their ability to offer local resolution at modest computational cost, unstructured grids are attractive in this context. However, unstructured grids suffer from drawbacks such as high computational overhead and inefficient generation schemes. Here we present a scheme for generating unstructured grids on the surface of the sphere that overcomes these limitations. We also show how the scheme can be easily used to allow efficient domain decomposition for parallel computations. The surface of the sphere is covered with a spherical spiral, which is used to provide an underlying structure for the grid. The spiral is populated by nodes, which are then connected using an advancing front technique to generate near-equilateral spherical triangular elements. Methods for producing local grid refinement by adjusting the pitch of the spherical spiral are discussed, as is the extension of the method to the case of coupled pressure-velocity solvers. The same general idea of a spherical spiral also serves as the starting point for an algorithm to subdivide the grid into subdomains for parallel computation. The resulting unstructured grids are generally of very high quality: in uniform grids, 99.4% of the elements have areas between 90 and 107% of the mean element area, and 99.8% of the edges have lengths between 84 and 132% of the mean edge length. The quality of the grids increases with mesh density. Partitioning of the nodes and elements produces wellbalanced and compact subdomains, with a maximum load imbalance that is small and rises gradually with number of subdomains. The proposed scheme produces grids that combine the benefits of an unstructured mesh with the structure conferred by the underlying spherical spiral. For example, this underlying structure greatly