A class of active flux schemes is developed and employed to solve model problems for the linear advection equation and non-linear Burgers equation. The active flux schemes treat the edge values, and hence the fluxes, as independent variables, doubling the degrees of freedom available to describe the solution without enlarging the stencil. Schemes up to third order accurate are explored. Oscillations generated by the higher-order schemes are controlled through the use of a characteristic-based limiter that preserves true extrema in the solution without excessive clipping.