Cryptocurrencies are distributed systems that allow exchanges of native (and non-) tokens between participants. The availability of the complete historical bookkeeping opens up an unprecedented possibility: that of understanding the evolution of a cryptocurrency's network structure while gaining useful insights into the relationships between users' behavior and cryptocurrency pricing in exchange markets. In this article we review some recent results concerning the structural properties of the Bitcoin Transaction Networks, a generic name referring to a set of three different constructs: the Bitcoin Address Network, the Bitcoin User Network, and the Bitcoin Lightning Network. The picture that emerges is of a system growing over time, which becomes increasingly sparse and whose mesoscopic structural organization is characterized by the presence of an increasingly significant core-periphery structure. Such a peculiar topology is accompanied by a highly uneven distribution of bitcoins, a result suggesting that Bitcoin is becoming an increasingly centralized system at different levels.