Many infrastructure systems can be modeled as networks, where a set of nodes is connected via some edges. Such a formulation allows us to consider the resilience properties of the network including its ability to maintain connectivity under some level of failures. Here we present a review on prior and recent results about such resilience properties We consider the general connectivity patterns of basic networks and review applications to specific systems like traffic, climate, and physiology. We also present results on the possibility of repairing a network after some failures. Next, we present results on interdependent networks, where one network depends on another for some resource. A typical example is a communications network depending on a power grid (and vice versa). Interdependent networks are often characterized by abrupt transitions where a cascade leads the network to collapse suddenly. We review possible methods of preventing these cascades such as reducing the level of interdependence, reinforcing some nodes, and other methods. Finally, we show how spatially embedded networks have unique properties such as extreme vulnerability in interdependent networks, metastable properties under localized attacks, and cascades under overload failures. Overall the results here provide possible methods and understanding of how to improve the resilience of modern critical infrastructure.