The article analyses the peculiar features of the functioning of cryptocurrency exchanges. In this paper, cryptocurrency exchanges on traditional trading platforms, direct trading exchanges, cryptocurrency brokers and cryptocurrency funds are classified. The difference between centralized and decentralized cryptocurrency exchanges is characterized. It was found that the activities of cryptocurrency exchanges and mining are mostly unregulated at the state level, and regulation of illegal transactions with cryptocurrency does not exist in most countries. The positions concerning cryptocurrency exchanges of those foreign financial regulators that regulate their activity are investigated. The basic requirements that cryptocurrency exchanges must meet to be licensed to conduct such activities in the USA and Japan are analysed. The article highlights one of the important contemporary problems in the modern cryptocurrency market - the regulation in Ukraine of transactions using cryptocurrency and the activity of cryptocurrency exchanges. The author presents approaches for adaptation of the national legislation of Ukraine regarding cryptocurrencies.