  Active Flux is a recently developed numerical method for hyperbolic
conservation laws. Its classical degrees of freedom are cell averages and point
values at cell interfaces. These latter are shared between adjacent cells,
leading to a globally continuous reconstruction. The update of the point values
includes upwinding, but without solving a Riemann Problem. The update of the
cell average requires a flux quadrature at the cell interface, which can be
immediately performed using the point values. This paper explores different
extensions of Active Flux to arbitrarily high order of accuracy, while
maintaining the idea of global continuity. We propose to either increase the
stencil while keeping the same degrees of freedom, or to increase the number of
point values, or to include higher moments as new degrees of freedom. These
extensions have different properties, and reflect different views upon the
relation of Active Flux to the families of Finite Volume, Finite Difference and
Finite Element methods.
