Summary: We estimate the weights of optimal compression structures of several types. Minimum weights of columns having solid square or circular cross-sections are compared with those of similar metal-foam-filled tubes, hollow tubes and tubes whose walls are foam-core sandwiches. Similarly, the minimum (of near-minimum) weights of wide sandwich compression panels are studied along with those of hat-stiffened, solid-skin panels and panels in which the skins and stiffeners are themselves metal-foam-core sandwiches. In these studies, weight comparisons are made on the basis of appropriate structural indices, and compressive strength is the only failure criterion. These studies provide baseline comparisons that ignore other possible design constraints, such as longitudinal stiffness, minimum gage and cost.