Since it was invented in 1986, elliptic curve cryptography (ECC) has been studied widely in industry and academy from different perspectives. Some of these aspects include mathematical foundations, protocol design, curve generation, security proofs, point representation, algorithms for inherent arithmetic in the underlying algebraic structures, implementation strategies in both software and hardware, and attack models, among others. The main advantage of ECC is that shorter keys (less-memory requirements and faster field arithmetic operations) can be used if compared with other cryptosystems, which has made it the ideal choice for implementing public key cryptography in resource constrained devices, as the ones found in the envisioned applications of the Internet of Things, e.g., wireless sensors. In this application domain, lightweight cryptography has emerged as the required one because of the scarce computing resources and limited energy in devices. In this paper, we present a survey of ECC in the context of lightweight cryptography. The aim of this paper is to identify the criteria that make an ECC-based system lightweight and a viable solution for using in practical constrained applications. Representative works are systematically revised to determine the key aspects considered in ECC designs for lightweight realizations. As a result, this paper defines, for the first time, the concept and requirements for elliptic curve lightweight cryptography.