We will study Caratheodory convergence, which is the convergence theory for domains corresponding to uniform convergence on compact sets for analytic functions, and we will use this convergence to develop Loewner theory. This theory gives, for an injective analytic map f0 on D, a family ft, 0 ≤ t < ∞, of such maps, such that fs(D) ⊂ ft(D) for s ≤ t and ∪0≤t<∞ft(D) = C. We will see that ft can be described as the solution to a partial differential equation, the Loewner equation; this equation, first described in 1923, has played a crucial role in the development of geometric function theory since then.