In line with the rising demand for smarter solutions and embedded systems, Microelectromechanical systems (MEMS) have gained increasing importance for digital computing devices and Internet-of-Things (IoT) applications, most notably for mobile wearable devices. This achievement is driven by MEMS resonators' inherent properties such as simplicity, sensitivity, reliability, and low power consumption. Hence, they are being explored for ultra-low-power computing machines. Several fundamental digital logic gates, switching, and memory devices have been demonstrated based on MEMS microstructures' static and dynamic behavior. The interest of researchers in using MEMS resonators is due to seeking an alternative approach to circumvent the notable current leakage and power density problems of complementary metal-oxide-semiconductor (CMOS) technology. The continuous miniaturization of CMOS has increased the operating speed and reduces the size of the device. However, this has led to a relative increase in the leakage energy. This drawback in CMOS has renewed the interest of researchers in mechanical digital computations, which can be traced back to the work of Charles Babbage in 1822 on calculating engines. This dissertation presents axially-loaded and coupled-MEMS resonators investigations to demonstrate memory elements and different logic functions. The studies in this dissertation can be categorized majorly into three parts based on the implementation of logic functions using three techniques: electrothermal frequency tunability, electrostatic frequency modulations, and activation/deactivation of the resonant frequency. Firstly, the influence of the competing effects of initial curvature and axial loads on the mechanical behavior of MEMS resonator arches are investigated theoretically to predict the tunability of arches under axial loads. Then, the concept of electrothermal frequency tunability is used to demonstrate fundamental 2-bit logic gates. However, this concept consumes a considerable amount of energy due to the electrothermal technique. Next, the dynamic memory element and combinational logic functions are demonstrated using the concept of electrostatic frequency modulation. Though this approach is energy efficient compared to the electrothermal technique, it does not support the cascadability of MEMS resonator-based logic devices. Lastly, complex multifunctional logic gates are implemented based on selective modes activation and deactivation, resulting in significant improvement in energy efficiency and enabling cascadability of MEMS resonator-based logic devices.