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Electronic systems built on flexible plastic films and stretchable rubber sheets have attracted new applications in many emerging fields. Integration of high-speed electronics such as microwave power amplifiers and switches can extend the applications even further with wireless capabilities. As such, flexible and stretchable microwave electronics represent opportunities for future electronics where remote capabilities are desired. Here, we review advances in numerous types of microelectronic devices used for fast, flexible, and stretchable electronic devices, as well as flexible and stretchable passive elements and circuitries. We first introduce the challenges associated with design and fabrication, and the characteristics required for high-frequency operation of the devices on foreign substrates. Second, we review the recent efforts that were made utilizing different types of high-performance semiconductors, which are ideal for high-speed flexible and stretchable electronics, such as silicon, compound semiconductors, and 1-D and 2-D materials. Third, passive electronic components fabricated on such substrates, including inductors, capacitors, and transmission lines, are reviewed. Finally, we discuss the flexible and stretchable microwave electronics at the circuit level and review the recent advances in making numerous types of flexible and stretchable microwave circuits for diverse applications. [ABSTRACT FROM AUTHOR]