Summary: The wide applicability of zero-knowledge interactive proofs comes from the possibility of using these proofs as subroutines in cryptographic protocols. A basic question concerning this use is whether the (sequential and/or parallel) composition of zero-knowledge protocols is zero-knowledge too. We demonstrate the limitations of the composition of zero-knowledge protocols by proving that the original definition of zero-knowledge is not closed under sequential composition; and that even the strong formulations of zero-knowledge (e.g., black-box simulation) are not closed under parallel execution. We present lower bounds on the round complexity of zero-knowledge proofs, with significant implications for the parallelization of zero-knowledge protocols. We prove that three-round interactive proofs and constant-round Arthur-Merlin proofs that are black-box simulation zero-knowledge exist only for languages in BPP. In particular, it follows that the ``parallel versions'' of the first interactive proofs systems presented for quadratic residuosity, graph isomorphism, and any language in NP, are not black-box simulation zero-knowledge, unless the corresponding languages are in BPP. Whether these parallel versions constitute zero-knowledge proofs was an intriguing open questions arising from the early works on zero-knowledge. Other consequences are a proof of optimality for the round complexity of various known zero-knowledge protocols and the necessity of using secret coins in the design of ``parallelizable'' constant-round zero-knowledge proofs.