A differential algebra of finite type over a field $\mathbb{k}$ is a filtered algebra A, such that the associated graded algebra is finite over its center, and the center is a finitely generated $\mathbb{k}$-algebra. The prototypical example is the algebra of differential operators on a smooth affine variety, when $\text{char}\mathbb{k} = 0$. We study homological and geometric properties of differential algebras of finite type. The main results concern the rigid dualizing complex over such an algebra A: its existence, structure and variance properties. We also define and study perverse A-modules, and show how they are related to the Auslander property of the rigid dualizing complex of A.