In this paper, the author defines the Riemann and Lebesgue integrals (more precisely, the Riemann and Lebesgue \(\Delta\)-integrals and \(\nabla\)-integrals) on time scales and studies their properties and relationship. In particular, the author presents results concerning the lower and upper Darboux sums, the Riemann sums, the Riemann and Lebesgue integrals, and certain mean value results on time scales, which are needed in the proofs. This paper will be useful for anyone interested in the theory of dynamic equations on time scales (measure chains). Reviewer notes that Sections 1-5 are contained in the book [\textit{M. Bohner} and \textit{A. Peterson}, ``Advances in dynamic equations on time scales'' (2003; Zbl 1025.34001)] in Chapter~5 by M.~Bohner and the author, and Appendix~A is contained in the same book in Chapter~1 by M.~Bohner, the author, and A.~Peterson.