Bilinear maps developed from the Weil and Tate pairings over elliptic curves constitute a great achievement in cryptography. By using them, one-round three-party key exchange was possible, and practical cryptographic primitives were designed, such as identity-based encryption schemes and aggregate signatures. This paper starts with a short overview on bilinear maps and some of its most important applications to cryptography. Then, it moves to the problem of constructing multi-linear maps as an effort to design one-round multi-party key-exchange protocols as well as more specialized cryptographic primitives such as efficient broadcast encryptions and unique digital signatures.