A linear series is a vector space of sections of a line bundle, or a collection of divisors parameterized by a projective space, or yet a rational map to a projective space. The interplay between these characterizations is very fruitful. The question that concerns us is: As a variety degenerates, to what linear series degenerate? This is a question that was addressed by many people, most famously in the theory of curves, by Eisenbud and Harris (limit linear series) for curves of compact type and Harris and Mumford (admissible covers) for maps to the projective line. I will explain three related ways of understanding such degenerations in a much more general setting. They are born from ideas first introduced by Osserman, and connect to the theory of Mustafin varieties. The results presented are joint work with several people: Eduardo Vital (Bielefeld), Omid Amini (Paris-Saclay), Renan Santos and Felipe de Léon (IMPA).