A disconnected free surface consists of several separate pieces (connectivity components). The connectivity components and solid walls bound a connected liquid region. Generally, the stability problem for such equilibrium configurations cannot be simply reduced to a set of independent problems for each connectivity component. Rather, the set of connectivity components, the solid walls and the connected liquid region must be considered simultaneously. In this review, we outline particular properties of the stability of mechanical equilibrium of systems with disconnected free surfaces and summarize the results of previous work. Several open problems are also discussed.