In recent years, elliptic curve cryptography (ECC) has gained widespread exposure and acceptance, and has already      been included in many security standards. Engineering of ECC is a complex, interdisciplinary research field encompassing such fields as mathematics, computer science, and electrical engineering. In this paper, we survey ECC implementation issues as a prominent case study for the relatively new discipline of cryptographic engineering. In particular, we show that the requirements of efficiency and security considered at the implementation stage affect not only mere low-level, technological aspects but also, significantly, higher level choices, ranging from finite field arithmetic up to curve mathematics and protocols.