Security and privacy are among the growing concerns of a Vehicular Ad hoc Network (VANET) which requires a high degree of liability from its participants. In this dissertation, we address security, anonymity, and privacy challenges of VANETs in the light of the IEEE standards for vehicular communications. VANET provides a variety of road-safety and other applications through wireless devices installed in vehicles and roadside infrastructure. However, a vehicle must not be able to take advantage of its anonymity for any misbehavior like sending false messages or malicious updates to other vehicles or a roadside infrastructure. We use proxy signature, identity-based signature, and elliptic curve cryptosystems to provide authentication for infrastructure generated messages, and anonymous authentication for vehicle originated messages. Also, we investigate how some of the MAC-layer weaknesses may impair the security of a VANET. Our solutions are lightweight, bandwidth friendly and compatible to the current standards of vehicular communications.