Gene expression is regulated by complex networks of interactions between RNAs and proteins. Proteins that interact with RNA have been traditionally viewed as either specific or non-specific; specific proteins interact preferentially with defined RNA sequence or structure motifs, whereas non-specific proteins interact with RNA sites devoid of such characteristics. Recent studies indicate that the binary “specific vs. non-specific” classification is insufficient to describe the full spectrum of RNA–protein interactions. Here, we review new methods that enable quantitative measurements of protein binding to large numbers of RNA variants, and the concepts aimed as describing resulting binding spectra: affinity distributions, comprehensive binding models and free energy landscapes. We discuss how these new methodologies and associated concepts enable work towards inclusive, quantitative models for specific and non-specific RNA–protein interactions.