Systems of language equations of the form X_i = φ_i(X₁,..., X_n) (1 ≤ t ≤ n) are studied, in which every φ_i may contain the operations of concatenation and complementation. The properties of having solutions and of having a unique solution are given mathematical characterizations. As decision problems, the former is NP-complete, while the latter is in co-RE and its decidability remains, in general, open. Uniqueness becomes decidable for a unary alphabet, where it is US-complete, and in the case of linear concatenation, where it is L-complete. The position of the languages defined by these equations in the hierarchy of language families is established.