Cryptocurrencies are distributed systems that allow exchanges of native (and non-) tokens among participants. The complete historical bookkeeping and its wide availability opens up an unprecedented possibility, i.e., that of understanding the evolution of their network structure while gaining useful insight on the relationships between user behaviour and cryptocurrency pricing in exchange markets. In this contribution we review some of the most recent results concerning the structural properties of Bitcoin Transaction Networks, a generic name referring to a set of different constructs: the Bitcoin Address Network, the Bitcoin User Network and the Bitcoin Lightning Network. A common picture that emerges out of analysing them all is that of a system growing over time, which becomes increasingly sparse, and whose structural organization at the mesoscopic level is characterised by the presence of a statistically-significant core-periphery structure. Such a peculiar topology is matched by a highly unequal distribution of bitcoins, a result suggesting that Bitcoin is becoming an increasingly centralised system at different levels.