AbstractIn this article, we present a novel meshfree framework for fluid flow simulations on arbitrarily curved surfaces. First, we introduce a new meshfree Lagrangian framework to model flow on surfaces. Meshfree points or particles, which are used to discretize the domain, move in a Lagrangian sense along the given surface. This is done without discretizing the bulk around the surface, without parametrizing the surface, and without a background mesh. A key novelty that is introduced is the handling of flow with evolving free boundaries on a curved surface. The use of this framework to model flow on moving and deforming surfaces is also introduced. Then, we present the application of this framework to solve fluid flow problems defined on surfaces numerically. In combination with a meshfree generalized finite difference method (GFDM), we introduce a strong form meshfree collocation scheme to solve the Navier–Stokes equations posed on manifolds. Benchmark examples are proposed to validate the Lagrangian framework and the surface Navier–Stokes equations with the presence of free boundaries.