An overview of various integrals is given which can be defined on arbitrary monotone set functions vanishing in the empty set (called here monotone measures). Our survey not only includes the Choquet integral [G. Choquet, Ann. Inst. Fourier (Grenoble) 5 (1954)], the Shilkret integral [N. Shilkret, Indag. Math. 33 (1971)] and the Sugeno integral [M. Sugeno, PhD thesis, Tokyo Institute of Technology (1974)] and some of their properties, but also some more general and more recent concepts as universal integrals [E. P. Klement, R. Mesiar, and E. Pap, IEEE Trans. Fuzzy Systems 18 (2010)] and decomposition integrals [Y. Even and E. Lehrer, Econom. Theory 56 (2014)], together with some of their properties, such as integral inequalities and convergence theorems.