In this paper the authors consider a systematic treatment of multilinear Calderón-Zygmund operators introduced earlier in the papers of Coifman and Meyer and of M. Lacey and C. Thiele [Ann. Math. (2) 146, 693-724 (1997; Zbl 0914.46034); ibid. 149, 475-496 (1999; Zbl 0934.42012)]. The first main result reads as follows: Let -linear operators be for which there is a function defined away from the diagonal in satisfying
whenever and . Let be given numbers with . Suppose that maps into if or if . Then for any such that , extends to a bounded map from into if all and into if some . If some , should be replaced by . Moreover, extends to a bounded map from to BMO. Next, the authors obtain the version of the multilinear T1 theorem by G. David and J.-L. Journé [Ann. Math. (2) 120, 371-397 (1984; Zbl 0567.47025)]. It is proved that if and , are bounded subsets of BMO, then has a bounded extension from into if . Here th transpose of is defined via
for all , in . This multilinear Calderón-Zygmund theory is applied to obtain some new continuity results for multilinear translation invariant operators, multlinear pseudodifferential operators, and multilinear multipliers.