His thesis concerned deformations of complex analytic spaces.
Subsequently, he became more interested in the work of Pierre Fatou and Gaston Julia and made significant contributions to the fields of analytic geometry and dynamical systems.
Together with his former student John H.
Hubbard, he launched a new subject, and a new school, studying properties of iterated quadratic complex mappings.
They made important mathematical contributions in this field of complex dynamics, including a study of the Mandelbrot set.
One of their most fundamental results is that the Mandelbrot set is connected; perhaps most important is their theory of renormalization of (polynomial-like maps).
The Douady rabbit, a quadratic filled Julia set, is named after him.
Douady taught at the University of Nice and was a Professor at the Paris-Sud 11 University, Orsay.