At the turning of the millennium, a switch in computing technology is forecasted and looked for. Two main directions of research, both based on quite unconventional ideas are most promising - quantum computing and molecular computing. In the last few years, both of these methods have been intensely investigated. The present book is the first "friendly" presentation of basic ideas in these exciting areas. The style is rigorous, but without entering into excessive technicalities. Equal attention is paid to the main practical results reported so far and the main theoretical developments. The book is written for the educated layman and is self-contained, including all the necessary facts from mathematics, computer science, biology and quantum mechanics.