We consider a membrane that adheres both weakly and strongly to a geometrically structured substrate. The interaction potential is assumed to be local, via the Deryagin approximation, and harmonic. Consequently, we can analytically describe a variety of different geometries; such as, smooth substrates interrupted by an isolated cylindrical pit, a single elongated trench, or a periodic array of trenches. We present more general expressions for the adhesion energy and membrane configuration in Fourier space and find that, compared with planar surfaces, the adhesion energy decreases. We also highlight the possibility of overshoots occurring in the membrane profile and look at its degree of penetration into surface indentations.