Configuration space integrals can be viewed as generalizations of Gauss’s linking integral. One application of them is to produce cohomology classes in embedding spaces, including all Vassiliev invariants of knots and links. For spaces of embeddings of 1-manifolds in higher-dimensional Euclidean spaces, they produce nontrivial "Vassiliev cohomology classes" in arbitrarily high degrees. For spaces of braids, they can be explicitly related to Chen’s iterated integrals. I will give the basic definitions and an overview of these results. I will also describe a topological reformulation of them, which yields rationality results for the Vassiliev classes and provides a potential avenue for seeking nontrivial torsion classes.