The use of marine controlled-source electromagnetic EM (CSEM) sounding to detect thin resistive layers at depths below the seafloor has been exploited recently to assess the resistivity of potential hydrocarbon reservoirs before drilling. We examine the sensitivity of the CSEM method to such layers with forward and inverse modeling in one and three dimensions. The 3D modeling demonstrates that if both source and receivers are over a tabular 3D target, 1D modeling predicts the observed response to very high accuracy. Experimental design can thus be based on 1D analysis in which hundreds of range and frequency combinations can be computed to find the optimal survey parameters for a given target structure. Modeling in three dimensions shows that the vertical electric-field response is largest over the edges of a 3D target. The 3D modeling also suggests that a target body needs to have a diameter twice the burial depth to be reliably seen by CSEM sounding. A simple air-wave model (energy propagating from source to receiver via the atmosphere) allows the effects of the target layer and atmosphere to be separated and shows where sensitivity to the target is diminished or lost because of finite water depth as a function of range, frequency, and seafloor resistivity. Unlike DC resistivity sounding, the marine CSEM method is not completely T-equivalent and, in principle, can resolve resistivity and thickness separately. Smooth inversion provides an estimate of the method's resolving power and highlights the fact that although the radial CSEM fields contain most of the sensitivity to the thin resistive target, inverted alone they produce only increasing resistivity with depth. Inclusion of the radial mode CSEM data forces the recovery of the thin resistor, but magnetotelluric data can be used more effectively to achieve the same result.