The principal advances in the design and construction, as well as the static, vibrational, and buckling analysis of thin-walled structures and buildings in the shape of general and axisymmetric ellipsoidal shells are summarized in this review. These shells are particularly useful as internally pressurized vessels or as heads and bottoms of cylindrical tanks and vessels. Reinforced concrete and structural steel domes of buildings, air-supported rubber-fabric shells, and underwater pressure vessels are also made in the form of ellipsoidal, shells. Knowing the geometry of ellipsoids, one can solve various problems in physics, optics, and so on. Basic results of theoretical and experimental investigations of the stress-strain state, buckling, and natural and forced vibrations contained in 209 references are presented in the review. The influence of temperature on the stress-strain state of the shells in question is also discussed. Some parts of the review are also devoted to an analysis of the literature on the stress-strain state of ellipsoidal and torispherical heads of pressure vessels with openings.