Cryptocurrencies are an inevitable product of the information technology of our time and have already become a significant participant in the world's financial markets. The striking development of the cryptocurrency market over the last few years – from being entirely peripheral to capitalizing at the level of an intermediate-size stock exchange – provides a unique opportunity to observe its evolution in a short period. The availability of high-frequency data allows conducting advanced statistical analysis of fluctuations on cryptocurrency exchanges right from their birth up to the present day. This opens a window that allows quantifying the evolutionary changes in the complexity characteristics which accompany market emergence and maturation. This presentation aims to review the properties of the cryptocurrency market and the associated phenomena. The aim is to clarify to what extent, after such an impetuous development, the characteristics of the complexity of exchange rates on the cryptocurrency market have become similar to traditional and mature markets, such as stocks, bonds, commodities or currencies. The stylized facts revealed - like the so-called ‘‘fat tails’’ of the returns distribution, volatility clustering, the ‘‘long memory’’, strong stochasticity alongside non-linear correlations, persistence, and the effects resembling fractality and even multifractality - may indicate direction of the future development of cryptocurrency market.