The Stiefel manifold is the set of orthonormal bases for $k$-planes in an $n$-dimensional space. We compute its degree as an algebraic variety in the set of $k$-by-$n$ matrices using techniques from classical algebraic geometry, representation theory, and combinatorics. We give an interpretation of this degree in terms of non-intersecting lattice paths. This is joint work with Fulvio Gesmundo.