Gaussian curvature in codimension > 1

The Gaussian curvature $K$ is a fundamental geometric quantity discovered by Gauss in the case of surfaces embedded in $\mathbb{R}^3$. One can naturally extend the definition of the Gaussian curvature to arbitrary submanifolds of $\mathbb{R}^k$ so that the extrinsic interpretation of $K$, the Theorema Egregium and the Gauss-Bonnet Theorem still hold. We give a concise exposition of these classical facts.