Active Flux method, first introduced by T. Eymann and P. Roe [1], is a well-established algorithm to solve linear and non-linear 2D and 3D hyperbolic PDEs in Cartesian coordinates. The main idea of the scheme is to separate variables into 2 classes: cell averages in the mesh cells and point values in the nodes and edges of the mesh. The scheme for the cell averages is constructed based on the finite volume method for the equations in their conservative (divergent) form. The scheme for point values typically uses a non-divergent form of equations, approximating the Jacobians and gradients of variables in the cells surrounding the point value. In this talk, we introduce the novel third-order Active Flux method for hyperbolic equations on Riemannian manifolds. For the mesh, we use triangulation of the surface with spherical triangles, where every triangle has its own local projection to a plane. The method for cell averages is carefully constructed to be geometry-compatible, i.e. so that numerical flux always lies in the tangent space of the manifold. The flux vector is split into geometry part and variable part. This allows to approximate the contour integral of the flux using the tangent (to the contour) vectors instead of normals, which leads to a consistent scheme. For the point values, we use a non-divergent form of equations, which doesn’t contain any Christoffel symbols. To approximate the gradients in each spherical triangle, a novel quasi-polynomial reconstruction of variables is used. Following the works on PAMPA method on a plane [2,3], we introduce first-order schemes for cell averages and point values. Rankine–Hugoniot conditions on manifolds are derived and used to construct Lax-Friedrichs like methods. The first and third-order schemes are blended together to obtain a bound-preserving scheme. Some tests for linear transport, acoustic and shallow water equations are provided.