Suction caissons, also known as suction anchors, are commonly used as part of offshore mooring systems. Loading is applied to the caisson at the mooring attachment point, the location of which can be optimised for maximum holding capacity. In this paper, plane strain and three-dimensional finite element limit analyses are used to find the capacity of suction caissons that are loaded horizontally. Lower and upper bounds on the capacity are found, with adaptive mesh refinement used to reduce the bound gap in successive iterations of the solution. The influences of the caisson length-to-diameter aspect ratio, the soil strength profile, and the load location are considered. The soil is modelled as a rigid-plastic von Mises material with either constant undrained shear strength or strength that increases linearly with depth. Capacity charts showing the variation in caisson capacity with load location are presented, and failure mechanisms are discussed.