Overlay networks are widely used to deploy functionality at edge nodes without changing network routers. Each node in an overlay network maintains pointers to a set of neighbor nodes. These pointers are used both to maintain the overlay and to implement application functionality, for example, to locate content stored by overlay nodes. If an attacker controls a large fraction of the neighbors of correct nodes, it can "eclipse" correct nodes and prevent correct overlay operation. This Eclipse attack is more general than the Sybil attack. Attackers can use a Sybil attack to launch an Eclipse attack by inventing a large number of seemingly distinct overlay nodes. However, defenses against Sybil attacks do not prevent Eclipse attacks because attackers may manipulate the overlay maintenance algorithm to mount an Eclipse attack. This paper discusses the impact of the Eclipse attack on several types of overlay and it proposes a novel defense that prevents the attack by bounding the degree of overlay nodes. Our defense can be applied to any overlay and it enables secure implementations of overlay optimizations that choose neighbors according to metrics like proximity. We present preliminary results that demonstrate the importance of defending against the Eclipse attack and show that our defense is effective.