Notions and applications of algorithmic randomness

Algorithmic randomness uses computability theory to define notions of randomness for infinite objects such as infinite binary sequences. The different possible definitions lead to a hierarchy of randomness notions. In this thesis we study this hierarchy, focussing in particular on Martin-Lof randomness, computable randomness and related notions. Understanding the relative strength of the different notions is a main objective. We look at proving implications where they exists (Chapter 3), as well as separating notions when the are not equivalent (Chapter 4). We also apply our knowledge about randomness to solve several questions about provability in axiomatic theories like Peano arithmetic (Chapter 5)