The focus of this correspondence is on nonlinear characteristics of cryptographic Boolean functions. First, we introduce the notion of plateaued functions that have many cryptographically desirable properties. Second, we establish a sequence of strengthened inequalities on some of the most important nonlinearity criteria, including nonlinearity, avalanche, and correlation immunity, and prove that critical cases of the inequalities coincide with characterizations of plateaued functions.We then proceed to prove that plateaued functions include as a proper subset all partially bent functions that were introduced earlier by Claude Carlet. This solves an interesting problem that arises naturally from previously known results on partially bent functions. In addition, we construct plateaued, but not partially bent, functions that have many properties useful in cryptography.