The objective of this work is to solve two real-world superresolution problems of fundamental importance: extending the resolutions of non-diffraction-limited imaging systems, and estimating soil moisture globally at high spatiotemporal resolution. We approach these problems as instances of multi-measurement superresolution and single-image superresolution, and develop specialized methods tailored to the specifics of each problem. To solve the first problem, we augment a conventional imaging system with a programmable mask and defocused lens, allowing us to capture superresolved images beyond the resolutions of both mask and sensor by factors greater than 4x without the use of mechanical motion or an image model. To solve the second problem, we develop a robust method for enhancing the spatial resolution of soil moisture retrievals from NASA's Soil Moisture Active Passive (SMAP) satellite by using low rank modeling to both recover missing values and implement a resolution enhancement method based on learning relationships between dominant high-resolution patterns and low-resolution covariates at all locations globally.