We review the history and current state of the art of diodes fabricated with organic semiconductors and other printable materials. In particular, we look at the integration of printed diodes into circuits and systems for applications, with particular emphasis on rectification, energy harvesting, and negative differential resistance (e.g. tunnel diodes). An overview of solution processed and printable organic and inorganic materials utilised in diodes is provided with an in depth analysis of their physics of operation. Furthermore, it is explained how the diverse array in which printed diodes can be implemented demonstrates their potential in the printed electronics industry.