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Abstract Atmospheric pressure plasmas intersecting with dielectric surfaces will often transition into surface ionization waves (SIWs). Several applications of these discharges are purposely configured to be SIWs. During propagation of an SIW over a dielectric surface, the plasma charges the surface while responding to changes in geometrical and electrical material properties. This is particularly important for non-planar surfaces where polarization of the dielectric results in local electric field enhancement. In this paper, we discuss results from computational investigations of negative and positive SIWs propagating over nonplanar dielectrics in three configurations—wavy surfaces, cuts through porous materials and water droplets on flat surfaces. We found that negative SIWs are particularly sensitive to the electric field enhancement that occurs at the crests of non-planar surfaces. The local increase in ionization rates by the electric field enhancement can result in the SIW detaching from the surface, which produces non-uniform plasma exposure of the surface. Positive SIWs tend to adhere to the surface to a greater degree. These trends indicate that treatment of pathogen containing droplets on surfaces may be best performed by positive SIWs. The same principles apply to the surfaces cut through pores. Buried pores with small openings to the SIW may be filled by plasma by either flow of plasma into the pore (large opening) or initiated by photoionization (small opening), depending on the size of the opening compared to the Debye length.