We consider solving second order advection-diffusion PDEs on general computational meshes of polygons. We develop a general framework that includes both high order finite element and finite volume approximation techniques. Standard finite elements on polygonal elements are either not available or lose accuracy. We develop new finite elements that maintain accuracy on polygons by including undistorted polynomials defined directly on the element. They possess the minimum number of degrees of freedom subject to the accuracy and Sobolev space conformity. The direct serendipity finite elements approximate scalar functions (such as pressures, concentrations, and saturations), while the direct mixed finite elements approximate vector functions (such as velocities). Unfortunately, solutions to advective problems can develop shocks or steep fronts, and thereby lose Sobolev space conformity. We discuss the challenges of using finite volume weighted essentially non oscillatory (WENO) techniques on polygonal meshes. We also present a robust and efficient procedure for producing accurate stencil polynomial approximations. We develop a new multilevel WENO reconstruction with adaptive order that combines stencil polynomials. The nonlinear weighting biases the reconstruction away from both inaccurate oscillatory polynomials of high degree (i.e., those crossing a shock or steep front) and smooth polynomials of low degree, thereby selecting the smooth polynomial(s) of maximal degree of approximation. Extension of the framework to three dimensions is also discussed. Applications are given to tracer flow and Richards equation.