The Jacobi equation is most commonly used as a basis for comparison theorems in Riemannian geometry. It is well known that the Riccati equation may be used for this purpose as well [cf. e.g. \textit{M. Gromov}, Structures métriques pour les variétés riemanniennes (1981; Zbl 0509.53034)]. The present paper develops that approach systematically. This leads to elegant proofs of e.g. the Bishop-Gromov volume comparison theorem and similar more general results.