The blockchain technology promises innovation by moving away from conventional centralized architectures, where trust is placed in a small number of actors, to a decentralized environment where a collection of actors must work together to maintain consensus in the overall system. Blockchain offers security and pseudo-anonymity to its adopters, through the use of various cryptographic methods. While much attention has focused on creating new applications that make use of this technology, equal importance must be given to studying naturally occurring phenomena in existing blockchain ecosystems and mitigating their effects where harmful. In this dissertation, we develop a novel open-source log-to-file system that provides the ability to record information relevant to events as they take place in live blockchain networks. Specifically, our open-source software facilitates in-situ measurements on full nodes in the live Bitcoin and Bitcoin Cash blockchain networks. This measurement framework sheds new light on many phenomena that were previously unknown or scarcely studied. First, we examine the presence and impact of churn, namely nodes joining and leaving, on the behavior of the Bitcoin network. Our data analysis over a two-month period shows that a large number of Bitcoin nodes churn at least once. We perform statistical distribution fitting to this churn and emulate it in our measurement nodes to evaluate the impact of churn on the performance of the Bitcoin protocol. From our experiments, we find that blocks received by churning nodes experience as much as five times larger propagation delay than those received by non-churning nodes. We introduce and evaluate a novel synchronization scheme to mitigate such effects on the performance of the protocol. Our empirical evaluation shows that blocks received by churning nodes that synchronize their mempools with peers have roughly half the delay in propagation experienced by those that do not synchronize their mempools. We next evaluate and compare the performance of three block relay protocols, namely the default protocol, and the more recent compact block and Graphene protocols. This evaluation is conducted over full nodes running the Bitcoin Unlimited client (which is used in conjunction with the Bitcoin Cash network). We find that in most scenarios, the Graphene block relay protocol outperforms the other two in terms of the block propagation delay and the amount of total communication associated with block relay. An exception is when nodes churn frequently and spend a significant fraction of time off the network, in which case the compact block relay protocol performs best. In-depth analyses reveal subtle inefficiencies of the protocols. Thus, in the case of frequent churns, the Graphene block relay protocol performs as many as two extra round-trips of communication to recover information necessary to reconstruct blocks. Likewise, an inspection of the compact block relay protocol indicates that the full transactions included in the initial block message are either unnecessary or insufficient for the successful reconstruction of blocks. Finally, we investigate the occurrence of orphan transactions which are those whose parental income sources are missing at the time that they are processed. These transactions typically languish in a local buffer until they are evicted or all their parents are discovered, at which point they may be propagated further. Our data reveals that slightly less than half of orphan transactions end up being included in the blockchain. Surprisingly, orphan transactions tend to have fewer parents on average than non-orphan transactions, and their missing parents have a lower fee, a larger size, and a lower transaction fee per byte than all other received transactions. Moreover, the network overhead incurred by these orphan transactions can be significant when using the default orphan memory pool size (i.e., 100 transactions), although this overhead can be made negligible if the pool size is simply increased to 1,000 transactions. In summary, this dissertation demonstrates the importance of characterizing the inner behavior of the peer-to-peer network underlying a blockchain. While our results primarily focus on the Bitcoin network and its variants, this work provides foundations that should prove useful for studying and characterizing other blockchains.