A perturbation theory is developed to describe modon propagation over slowly varying topography. The theory is developed from the rigid-lid shallow-water equations on an infinite β-plane. Nonlinear hyperbolic equations are derived, based on the conservation of energy, enstrophy and vorticity, to describe the evolution of the slowly varying modon radius, translation speed and wavenumber for arbitrary finite-amplitude topography. To leading order, the modon is unaffected by meridional gradients in topography. Analytical perturbation solutions for the modon radius, translation speed and wavenumber are obtained for small-amplitude topography. The perturbations take the form of hyperbolic transients and a stationary component proportional to the topography. The solution predicts that as the modon moves into a region of shallower (deeper) fluid the modon radius increases (decreases), the translation speed decreases (increases) and the modon wavenumber decreases (increases). In addition, as the modon propagates into a region of shallower (deeper) fluid there is an amplification (diminishing) of the extrema in the streamfunction and vorticity fields. These properties suggest that the modon may be able to be topographically-captured and amplified, and thus may have application to the onset of atmospheric blocking. The general solution is applied to mid-latitude scales and a ridge-like topographic feature.