This paper studies the homotopy theory of algebras and homotopy algebras over an operad. It provides an exhaustive description of their higher homotopical properties using the more general notion of morphisms called infinity-morphisms. The method consists in using the operadic calculus to endow the category of coalgebras over the Koszul dual cooperad or the bar construction with a new type of model category structure, Quillen equivalent to that of algebras. We provide an explicit homotopy equivalence for infinity-morphisms, which gives a simple description of the homotopy category, and we endow the category of homotopy algebras with an infinity-category structure.