  Differential equations with the Painlev\'e property have been studied
extensively due to their appearance in many branches of mathematics and their
applicability in physics. Although a modern, differential algebraic treatment
of the order one equations appeared before, the connection with the classical
theory did not. Using techniques from algebraic geometry we provide the link
between the classical and the modern treatment, and with the help of
differential Galois theory a new classification is derived, both for
characteristic 0 and p.
