Accurate description of subsurface flow and transport properties and their spatial connectivity is critical for predicting and optimizing resource development efficiency. Geologic uncertainty, complex (multi-scale) spatial heterogeneity, and insufficient data complicate the description and characterization of subsurface physical properties. A common practice to reduce uncertainty and improve flow predictions is to use dynamic response data from scattered well locations to constrain (calibrate) high-resolution subsurface flow models. However, dynamic data from limited locations do not provide sufficient content to resolve high-resolution details in flow models. This discrepancy calls for effective descriptions of subsurface models that can be resolved from dynamic response data. In our research, we take a feature-based description approach that focuses on representing and estimating the large-scale connectivity in subsurface flow properties. Geologic formations consist of layers of rock deposits with similar properties across their horizontal extent, and hence exhibit strong spatial continuity and correlation. This prevalent geologic continuity enables parsimonious representations of the related properties in a properly designed transform domain. We develop various low-rank representation methods for effective subsurface description and inverse modeling applications. Our research in this area is inspired by signal/image processing, machine learning, and approximation theory. In particular, we are interested in sparse representation approaches that facilitate model recovery and calibration via sparse reconstruction algorithms. Some of our specific research topics include developing efficient application-driven parameterization techniques, designing spectral-domain compact representation methods, constructing learned sparse geologic dictionaries for describing and characterizing complex (non-Gaussian) geologic environments, and developing robust descriptions for selection and identification of consistent geologic scenario(s) based on dynamic flow and pressure data.