The unique superfluid properties which distinguish the A phase of $^3$He from other superfluids (He II, the B phase of $^3$He, and electrons in superconductors) are described. These properties result from the specific breaking of the invariance of the state under gauge transformations and rotations in orbital space, so that all the superfluid properties are related in a fundamental way to the dynamics of textures of the liquid-crystal-anisotropy vector. The flow of the superfluid component of the liquid, for example, is not a potential flow; its vortical part is related to textures. The relationship between the structure of the superfluid velocity and the nature of the symmetry breaking is analyzed in detail. Simple phenomenological considerations are used to derive an expression for the current and to explain the strange term in the current which is not present in the model of a Bose gas of molecules having a Cooper-pair structure in the A phase. The existence of a normal density at T = 0 and the nonlocal properties of the density of the intrinsic orbital angular momentum of the liquid are discussed. The coexistence of superfluid properties with a nonpotential superfluid flow and also with the vanishing of the critical Landau velocity for the creation of excitations is discussed. Conditions for the global and local stability of the superfluid flow are described. The changes in stability upon changes caused in the topological structure of the order parameter by external conditions are discussed. The relaxation dynamics of a superfluid flow is analyzed. An instanton mechanism for phase slippage is analyzed; this mechanism leads to observable oscillations analogous to the ac Josephson effect in superconductors. Principles for constructing a generalized two-fluid Landau hydrodynamics taking an orbital variable into account are outlined. Continuous and singular vortex structures which arise upon rotation are discussed, as are their topology, phase transitions between them, and some other topologically nontrivial structures: a vortex with a free end, a boojum, and an instanton. The basic experiments in which the unique superfluid properties are observed are discussed. Some other existing or possible systems in which a nonstandard breaking of gauge invariance leads to unusual properties are cited; examples are a neutron star and a superconductor with the A-phase structure. In the latter medium there is a structure similar to a Dirac magnetic monopole. This structure is discussed.