The classical de Rham isomorphism between the singular cohomology of a smooth manifold $M$ and its de Rham cohomology is a well-known theorem. In this talk we will consider Riemannian manifolds that admit a certain triangulation $h: K \to M$, where $K$ is a simplicial complex in $\R^n$. We will establish the notion of $L_p$-norms for differential forms on $M$ as well as for simplicial cochains on $K$. These define two cochain complexes $L_p(K)$ and $L_p(M)$. Its cohomologies $H_p(K)$ and $H_p(M)$ are the $L_p$-cohomology of $M$. We will discuss a variant $H_p(K) \to H_p(M)$ of the de Rham isomorphism that holds in this particular setting. We will also sketch some other aspects of $L_p$-spaces. A full script is available at http://math.nikno.de/.