Biological membranes undergo thermal fluctuations affecting their curvature. These fluctuations are generally considered to decrease the experimentally observed mobility by increasing the membrane thickness or extending the geometric path. Taking the dynamic membrane fluctuations into account would require the treatment of changing surfaces. Instead, here, we restrain ourselves to work in the static membrane limit, which is an excellent approximation for structures such as the cristae in the endoplasmic reticulum (ER) or auditory outer hair cells. Another aspect of the classification of membrane surfaces is based on their curvature. The curvature of a surface can be completely described by two scalar fields, the Gaussian curvature K(r) and the mean curvature H(r). Caveolae and budding vesicles with nonzero Gaussian curvature have radii of few dozen to a hundred nanometers. Tubular structures such as those in the mitochondrion are examples of developable surfaces, that is, they only have non-vanishing mean curvature. The ER is also of great interest due to its complex membrane topology of folded membranes with highly curved regions, and due to being the site of lipid droplet biogenesis. Several studies employing continuum methods have investigated the effect of both mean and Gaussian curvature on biological membranes, resulting in two major conclusions.