  This work presents and studies Riccati equations over finite-dimensional
normed division algebras. We prove that a Riccati equation over a
finite-dimensional normed division algebra $A$ is a particular case of
conformal Riccati equation on a Euclidean space and it can be considered as a
curve in a Lie algebra of vector fields $V\simeq\mathfrak{so}(\dim A+1,1)$.
Previous results on known types of Riccati equations are recovered from a new
viewpoint. A new type of Riccati equations, the octonionic Riccati equations,
are extended to the octonionic projective line $\mathbb{O}{\rm P}^1$. As a new
physical application, quaternionic Riccati equations are applied to study
quaternionic Schr\"odinger equations on 1+1 dimensions.
