Let \(G\) be a Carnot group and let \(\mathcal{L}\) be its left-invariant sub-Laplacian. As a consequence of Hörmander's theorem, \(\mathcal{L}\) is hypoelliptic, that is, for every distribution \(u\) on \(G\), if \(\mathcal{L}u \in C^\infty\), then \(u \in C^\infty\). In this paper, the authors give a self-contained proof of the hypoellipticity of \(\mathcal{L}\). Hörmander's theorem, of course, is much more general: it applies to any operator of the form \(\mathcal{L} = \sum_{j=1}^q X_j^2 + X_0\), where \(X_0, X_1, \dots, X_q\) are vector fields satisfying a bracket-generating condition. A few different proofs have been given, but all are rather difficult. The authors give a brief history of the theorem, citing in particular [\textit{L. Hörmander}, Acta Math. 119, 147--171 (1967; Zbl 0156.10701)], [\textit{J. J. Kohn}, Proc. Sympos. Pure Math. 23, 61--69 (1973; Zbl 0262.35007)] and [\textit{P. Malliavin}, in: Proc. int. Symp. on stochastic differential equations, Kyoto 1976, 195--263 (1978; Zbl 0411.60060)]. So, in this paper, the authors give a new and simpler proof in the special case of the sub-Laplacian of a Carnot group in which the vector fields \(X_1, \dots, X_q\) are left-invariant and \(X_0=0\). The proof is based on subelliptic estimates for Sobolev spaces \(W^{k,p}_X\) defined in terms of the left-invariant vector fields \(X_1, \dots, X_q\), rather than the Euclidean derivatives \(\frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_N}\). A key technique is to study the interaction between the norms of \(W^{k,p}_X\) and the analogous spaces \(W^{k,p}_{X^R}\) defined using the right-invariant vector fields instead. The main regularity estimate, Theorem 3.4, bounds the \(W^{k,2}_{X^R}\) norm of \(f\) in terms of higher \(X^R\)-Sobolev norms of \(\mathcal{L} f\), under the hypothesis that \(f \in W^{1,2}_{X}\), and its proof occupies most of the paper. In particular, Theorem 3.4 implies that if \(f \in W^{1,2}_X\) and \(\mathcal{L} f \in C^\infty\), then \(f \in C^\infty\). The remainder of the paper (Section 4) is concerned with using a regularization argument to drop the hypothesis \(f \in W^{1,2}_X\), thus obtaining the full result for arbitrary distributions. As a general principle, the authors have focused on writing a clear exposition, using techniques as elementary as possible, rather than on obtaining the sharpest possible results.