We developed a comprehensive analytical, numerical, and experimental methodology to consider structural nonlinearity as a main design factor enabling to tailor mechanical resonances and achieve targeted performance. Motivated by the need to advance current capabilities of micro systems, our research has been focused on the implementation of intentional intrinsic nonlinearity in the design of micro resonators and proved that harnessing intentional strong nonlinearity enables exploiting various nonlinear phenomena, not attainable in linear settings, such as broadband resonances, dynamic instabilities, nonlinear hysteresis, and passive targeted energy transfers. Therefore, understanding the mechanisms leading to nonlinearity in such systems will eliminate obstacles to their further development and significantly enhance their performance. Due to their small size and low damping, these devices often exhibit significant nonlinearity and thus the operational range of these impressive applications shrinks. During the last decades, we have witnessed various micro systems revolutionized fundamental and applied science.