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Chern-Simons invariants of closed oriented Riemannian $3$-manifolds areintroduced and studied from the basics. Their first-order variation is theCotton tensor. The properties of the Cotton tensor: symmetry, conformalcovariance, trace- and divergence-freedom, are recovered as corollaries of theChern-Simons invariant. We prove that the Cotton tensor is the obstruction tolocal conformal flatness in dimension $3$. Finally we discuss the link betweenChern-Simons invariants, the eta invariant, and the central value of theSelberg zeta function. This is a survey paper without original results, basedon lecture notes for a doctoral course at the University of Bucharest, March2014.

The Cotton tensor and Chern-Simons invariants in dimension $3$: an introduction