We present a selection of topics relating to modeling, designing, and measuring EUV (Extreme Ultraviolet) photomasks, with implications for high-volume nanofabrication of integrated circuits. These EUV photomasks must be accurately designed, but rigorously modeling large domains is extremely computationally intensive; we introduce an approximateFresnel Double Scattering model which is 10,000x faster. This approximation can predict the trend of phase vs pitch, which is critical to designing EUV phase shift masks (PSMs). We also explore novel mask architectures to improve efficiency and contrast, such as an etched multilayer PSM (up to 6x throughput but restrictive applicability), aperiodic multilayers (up to +22% throughput and more general applicability), and multilayers with minimal propagation distance at certain angles (lower throughput but higher contrast with minimized 3D effects). Finally we explore computational metrology with EUV reflectometry, scatterometry, and imaging for probing the phase and amplitude response of an EUV mask, with experimental demonstrations at the Advanced Light Source synchrotron. We perform reflectometry experiments on 3 masks with different architectures to infer approximately 25 physical film parameters each. Another reflectometry application to contamination monitoring achieved single-picometer precision for thickness (3σ < 6pm) and sub-degree precision for phase (3σ < 0.2deg). We compare two implementations of phase scatterometry, either applying nonlinear optimization with approximate scattering, or linearizing the rigorous scattering relationship between intensity and phase; linearization is shown to generally be more accurate, but both methods have similar precision. We apply novel software and hardware for phase imaging, using PhaseLift convex phase retrieval, combined with a set of custom Zernike Phase Contrast (ZPC) zone plates. We perform hyperspectral ZPC phase imaging on 3 masks, where we see promising agreement with reflectometry in the trend of phase vs wavelength.