  We present a finite volume method that is applicable to hyperbolic PDEs
including spatially varying and semilinear nonconservative systems. The spatial
discretization, like that of the well-known Clawpack software, is based on
solving Riemann problems and calculating fluctuations (not fluxes). The
implementation employs weighted essentially non-oscillatory reconstruction in
space and strong stability preserving Runge-Kutta integration in time. The
method can be extended to arbitrarily high order of accuracy and allows a
well-balanced implementation for capturing solutions of balance laws near
steady state. This well-balancing is achieved through the $f$-wave Riemann
solver and a novel wave-slope WENO reconstruction procedure. The wide
applicability and advantageous properties of the method are demonstrated
through numerical examples, including problems in nonconservative form,
problems with spatially varying fluxes, and problems involving near-equilibrium
solutions of balance laws.
