Introduction: the historical context. My goal in this article, as it was in the lecture at the Logic Colloquium in Athens, is to survey the notion of independence of types, a fundamental tool in the area of stability theory, and see different ways in which it can be realized in a particular example of an unstable theory, the theory of algebraically closed valued fields. I thank the anonymous referee for many comments which have significantly improved this article. Of course, all remaining errors are my own.

The notion of independence was first formulated by Shelah [Sh] in the 1970's in the context of classification theory. The motivating problem was the following.

problem. Given a theory T and a cardinal λ > |T|, let I (T, λ) be the number of models of T of cardinality λ, up to isomorphism. What can the function I (T, λ) be?

The first observation is that there is the following fundamental dichotomy.

• Suppose T is unstable; that is, for every uncountable λ there is a parameter set A with |A| ≤ λ such that the number of types over A is 2λ. In this case, I (T, λ) = 2λ; the maximum possible value.

• Suppose T is stable. Then there are different possibilities for I (T, λ), so one can look for further conditions on the theory which will serve to determine the value that I (T, λ) takes.