Let \(E\) be a Dedekind complete Riesz space and \([a,b]\) a real interval. In the first part of the paper, the authors give a characterization of ``Riemann-integrable'' functions \(f:[a,b]\to E\) well known in the real-valued case. In the second part, they introduce an integral for real-valued functions with respect to an \(E\)-valued measure: A function \(f\) is integrable if there is an \(L^1\)-Cauchy sequence of simple functions \((o)\)-converging in measure to \(f\). This concept of integrability is compared with several other concepts of integrability.