Optical imaging has seen significant development over the last few decades thanks to the advances of light sources and innovative imaging modalities. New quests have opened for advanced optical system design and engineering that require comprehensive understanding of theory and extensive computer simulations. This dissertation discusses several selected topics in advanced optical design and engineering, including new optical design considerations for systems with ultrafast illumination, engineering of a reflective microscope system in the vacuum ultraviolet and a method to design apochromat and superachromat objectives. For optical systems with ultrafast illumination, a modified definition of Strehl ratio is proposed to quantify chromatic and temporal behavior of ultrafast laser pulses at the optical focus. A simple method to obtain approximate numerical solutions is given with the help of ray tracing software. Effects of monochromatic aberrations, material dispersion up to the second order and pupil aberrations are discussed. System engineering of an imaging microscope illuminated with hydrogen Lyman-? line at 121.6nm is discussed. Challenges of diamond turning fabrication technology are described. Alignment and testing procedures are presented. A correction phase plate design is proposed for improving the as-built system performance, and a sensitivity analysis is carried out. For future work, a new two-stage system design is proposed to address the limitations of the current system. A simple method to design apochromats is proposed, where an apochromat is formed by combining two achromatic doublets with proper scaling of the focal lengths. A scaling formula is derived to calculate the focal lengths of each component. The formula is developed for lenses in contact and remote lenses with different marginal ray heights