I will explain my recent work on iterated beta integrals with Minoru Hirose at Nagoya University. Iterated beta integrals are a type of iterated integrals on branched universal abelian coverings of the projective line, which commonly generalize hyperlogarithms and beta integrals. In addition to the properties that generalize those of hyperlogarithms and beta integrals, iterated beta integrals enjoy a new property that we call translation invariance. The translation invariance yields equalities between iterated integrals on translation-equivalent coverings of the projective line, and especially for genus-zero pair of translation-equivalent coverings, it yields non-trivial equalities between special values of hyperlogarithms. Such equalities include the famous 2-3-2 formula for multiple zeta values proved by Zagier, its analog for multiple t-values, the 2-1 formula for multiple zeta star values, and various other formulas of a similar kind. By classifying all genus-zero pairs, some new interesting cases popped up, one of which resolves a conjecture by Charlton concerning the area of a family of Lawson surfaces.