Ordinals and conservativity spectra of arithmetical theories

We will give a survey of the results on the role of ordinals and ordinal notation systems in the study of formal theories, as well as present some open problems in this area. The recently emerged notion of conservativity spectrum bears more information on a formal theory than its proof-theoretic ordinal. It leads to algebraic type of structures, associated with a given theory. This allows one to obtain a more refined classification of theories than using more traditional approaches.