Nowadays, integrated circuits (IC) are mostly implemented using Complementary Metal Oxide Semiconductor (CMOS) transistor technology. This technology has allowed the chip industry to shrink transistors and thus increase the device density, circuit complexity, operation speed, and computation power of the ICs. However, in recent years, the scaling of transistor has faced multiple roadblocks, which will eventually lead the scaling to an end as it approaches physical and economic limits. The dominance of sub-threshold leakage, which slows down the scaling of threshold voltage VTH and the supply voltage VDD, has resulted in high power density on chips. Furthermore, even widely popular solutions such as parallel and multi-core computing have not been able to fully address that problem. These drawbacks have overshadowed the benefits of transistor scaling. With the dawn of Internet of Things (IoT) era, the chip industry needs adjustments towards ultra-low-power circuits and systems. In this thesis, energy-efficient Micro-/Nano-electromechanical (M/NEM) relays are introduced, their non-leaking property and abrupt switch ON/OFF characteristics are studied, and designs and applications in the implementation of ultra-low-power integrated circuits and systems are explored. The proposed designs compose of core building blocks for any functional microprocessor, for instance, fundamental logic gates; arithmetic adder circuits; sequential latch and flip-flop circuits; input/output (I/O) interface data converters, including an analog-to-digital converter (ADC), and a digital-to-analog converter (DAC); system-level power management DC-DC converters and energy management power gating scheme. Another contribution of this thesis is the study of device non-ideality and variations in terms of functionality of circuits. We have thoroughly investigated energy-efficient approximate computing with non-ideal transistors and relays for the next generation of ultra-low-power VLSI systems.