This paper is a survey of a part of the program of reverse mathematics (Simpson’s book [{\it S. G. Simpson}, Subsystems of second order arithmetic. Springer, Berlin (1999; Zbl 0909.03048)] is the main reference). The paper surveys all reverse mathematics results about countable well-orderings, which represent the natural coding of countable ordinals in subsystems of second-order arithmetics. Material touched upon includes: definitions of well-ordering, comparability issues and existence of suprema, addition, multiplication and exponentiation, Cantor’s normal form theorem. It does not include results related to ordinal notations or to the proof-theoretic ordinals of the various subsystems. The paper contains almost no proofs. The results are mostly contained in papers written over the span of more than a decade (in many cases by the author). This survey contains all the appropriate references and is very useful to find a quick answer to questions regarding the axiomatic strength of various statements about well-orderings. In most cases the answer supports the claim that $\text{\bf ATR}_0$ is the weakest system allowing a good theory of well-orderings.
Reviewer: Alberto Marcone (Udine)