This dissertation presents a novel modeling framework for understanding the dynamics of peer-to-peer (P2P) networks under churn (i.e., random user arrival/departure) and designing systems more resilient against node failure. The proposed models are applicable to general distributed systems under a variety of conditions on graph construction and user lifetimes. The foundation of this work is a new churn model that describes user arrival and departure as a superposition of many periodic (renewal) processes. It not only allows general (non-exponential) user lifetime distributions, but also captures heterogeneous behavior of peers. We utilize this model to analyze link dynamics and the ability of the system to stay connected under churn. Our results offers exact computation of user-isolation and graph-partitioning probabilities for any monotone lifetime distribution, including heavy-tailed cases found in real systems. We also propose an age-proportional random-walk algorithm for creating links in unstructured P2P networks that achieves zero isolation probability as system size becomes infinite. We additionally obtain many insightful results on the transient distribution of in-degree, edge arrival process, system size, and lifetimes of live users as simple functions of the aggregate lifetime distribution. The second half of this work studies churn in structured P2P networks that are usually built upon distributed hash tables (DHTs). Users in DHTs maintain two types of neighbor sets: routing tables and successor/leaf sets. The former tables determine link lifetimes and routing performance of the system, while the latter are built for ensuring DHT consistency and connectivity. Our first result in this area proves that robustness of DHTs is mainly determined by zone size of selected neighbors, which leads us to propose a min-zone algorithm that significantly reduces link churn in DHTs. Our second result uses the Chen-Stein method to understand concurrent failures among strongly dependent successor sets of many DHTs and finds an optimal stabilization strategy for keeping Chord connected under churn.