In this chapter, we reconstruct some of the traditional Fuchsian theory of regular singular points of meromorphic differential equations. We first introduce a quantitative measure of the irregularity of a singular point. We then recall how in the case of a regular singularity (i.e., a singularity with irregularity equal to zero), one has an algebraic interpretation of the eigenvalues of the monodromy operator around the singular point, using the notion of exponents. We then describe how to compute formal solutions of meromorphic differential equations and sketch the proof of Fuchs’s theorem: the formal solutions of a regular meromorphic differential equation all converge in some disc. We finally establish the Turrittin–Levelt–Hukuhara decomposition theorem, which gives a decomposition of an arbitrary formal differential module analogous to the eigenspace decomposition of a complex linear transformation. The search for an appropriate p-adic analogue of this result will lead us in Part V to the p-adic local monodromy theorem.