This paper surveys a variety of approaches to integration theory which have arisen in the context of continuous domain theory. Especially the author focuses attention on developing a constructive approach to Riemann-like integration. Section 2 contains some definitions and basic facts on a valuation. Section 3 treats the horizontal integral, which integrates an upper measurable, non-negative function with respect to a valuation. It is also shown that the integral can be approximated by the integrals of the canonical sequence of simple functions. Section 4 contains some properties of the S-integral, which is the integral for a non-negative finitely additive measure on an algebra of subsets of a non-empty set. Section 5 collects the results concerning the continuity and the uniqueness of the bilinear mapping defined by the horizontal integral. Section 6 contains some results which provide computational schemes for the S-integral in the case where a valuation is given by a directed supremum of known simple ones. The paper ends in Section 7 with giving an outline of the theory of the Riemann-Edalat integral.