Felix Klein, when discussing how the popularity of areas in mathematics rises and falls, mentions that in his youth Abelian functions were at the summit of mathematics, and that later on their popularity plummeted. I could hardly find anything on the net on Abelian functions and Wikipedia thinks that they are barely worth mentioning. What did the 19th century theory of these functions consist of, and why was it so important?

2$\begingroup$ I know this isn't quite what you asked, but the study of Abelian varieties  which is the modern descendant of Abelian function theory  is still extremely active and central to algebraic geometry, number theory... $\endgroup$– Donu ArapuraCommented May 31, 2010 at 12:02

3$\begingroup$ To make the relationship precise, an Abelian function would be the same thing as a meromorphic function on an Abelian variety, in the same way that an elliptic function is a function on an elliptic curve. $\endgroup$– Donu ArapuraCommented May 31, 2010 at 12:12

1$\begingroup$ I wonder what book of Klein were you refering to. He gave a good account in "Lectures on the development of mathematics in the 19th century". $\endgroup$– Victor ProtsakCommented May 31, 2010 at 14:12

6$\begingroup$ I think that, looking back from our vantage point, it is fair to interpret Klein's "Abelian functions" as referring to the subjects that would now be called "algebraic geometry" and "automorphic forms". These continue to be major parts of mathematics, even if, at the turn of the last century, they seemed to Klein to be waning in popularity. (Indeed, he laments that people are turning away from Abelian functions to axiomatics; but these axiomatics later played an enormous role in the development of modern algebraic geometry.) $\endgroup$– EmertonCommented Jun 1, 2010 at 2:24

$\begingroup$ @DonuArapura are the abelian functions the only meromorphic functions that can be defined on an abelian variety ? $\endgroup$– PonceCommented Dec 11, 2020 at 21:09
2 Answers
For a really detailed answer to your question, see The Legacy of Niels Henrik Abel, edited by O.A. Laudal and R. Piene (Springer 2004). In particular, there is a long introductory article by Christian Houzel, most of which can be viewed here.
Addendum. The complete article "The Work of Niels Henrik Abel" by Christian Houzel may be found here.
Recall C. L. Siegel's rant, about the modern theory of abelian functions not having any functions in it.
From a point of view that would have made sense to Weierstrass, mathematics has "addition theorems", such as one first meets for sin and cos. One achievement of the 19th century was to classify these addition theorems, in "several variables" (which were of course complex variables for those guys), which were "algebraic". Scare quotes not too serious: this is one origin of today's theory of algebraic groups. To a first approximation, the theory of projective algebraic groups is the theory of abelian functions. Riemann had in fact written down enough theta functions so that abelian functions could be expressed in terms of their quotients. This was all a big and muchsought general framework extending the elliptic functions. The reason to do that was that, for example, indefinite integrals of square roots of cubics and quartics should have their excellent theory extended to quintics and beyond.
It turns out that abelian functions in at least two variables is not quite the formulafest that the theory of elliptic functions is. Not for want of trying. We have the conceptual and geometric framework in the theory of abelian varieties. There are distinguished mathematicians who will tell you not to write down explicit functions. There is plenty of work, though, for example stimulated by soliton theory, explicit computation on curves of small genus, and so on.