Microfluidics has emerged as a transformative technology for precise control and manipulation of fluids, particles, and cells, revolutionizing various fields including biology, chemistry, medicine, and biotechnology. Inertial microfluidics, a subset of microfluidics, has attracted considerable attention due to its passive and label-free processing capabilities for large sample volumes at high flow rates, offering notable advantages in cell sorting, particle separation, and focusing. Moreover, the integration of oscillatory flow in microfluidic systems has unlocked novel opportunities to explore particle behavior, particularly in the context of inertial effects. Oscillatory microfluidics enables particles to engage in a continuous back-and-forth motion, emulating an infinite channel and facilitating efficient particle focusing and separation within shorter channel lengths compared to continuous unidirectional flow. This integration of inertial microfluidics and oscillatory flow holds great promise for precise particle manipulation and separation, with broad implications in biology, and medicine. Ongoing research endeavors aim to deepen our understanding of fluid-particle interactions, optimize channel geometries, and develop innovative strategies to enhance particle manipulation and separation efficiency. This dissertation presents two pivotal chapters that contribute to the advancement of particle separation techniques. Chapter two introduces a microfluidic device for size-based particle separation in a continuous inertial regime, while chapter three investigates the longitudinal separation of particles in oscillatory inertial microfluidics, unveiling insights into particle motion dynamics and paving the way for novel approaches to particle manipulation and longitudinal separation. These advancements hold immense potential for diverse applications, spanning from biomedical research to diagnostics, as continued research and development propel particle manipulation techniques forward.