This paper addresses the problem of resilient consensus in the presence of misbehaving nodes. Although it is typical to assume knowledge of at least some nonlocal information when studying secure and fault-tolerant consensus algorithms, this assumption is not suitable for large-scale dynamic networks. To remedy this, we emphasize the use of local strategies to deal with resilience to security breaches. We study a consensus protocol that uses only local information and we consider worst-case security breaches, where the compromised nodes have full knowledge of the network and the intentions of the other nodes. We provide necessary and sufficient conditions for the normal nodes to reach consensus despite the influence of the malicious nodes under different threat assumptions. These conditions are stated in terms of a novel graph-theoretic property referred to as network robustness.