In this memoir the authors revisit Almgren’s theory of $Q$-valued functions, which are functions taking values in the space $/mathcal{A}_Q(/mathbb{R}^{n})$ of unordered $Q$-tuples of points in $/mathbb{R}^{n}$. In particular, the authors: give shorter versions of Almgren’s proofs of the existence of $/mathrm{Dir}$-minimizing $Q$-valued functions, of their Hoelder regularity, and of the dimension estimate of their singular set; propose an alternative, intrinsic approach to these results, not relying on Almgren’s bi Lipschitz embedding $/xi: /mathcal{A}_Q(/mathbb{R}^{n})/to/mathbb{R}^{N(Q, n)}$; improve upon the estimate of the singular set of planar $/mathrm{D}$-minimizing functions by showing that it consists of isolated points.