The following are a (now obsolete) collection of notes on very large cardinals. The right resource now is this article on arxiv, that it is still a work in progress and will be updated regurarly. When finished, it should be an introduction to Very Large Cardinals, from I3 to hypotheses beyond I0. It is still a draft, so it is possible that I left some typos or grammar errors, but the mathematic behind it is solid. Since their introductory value, I wrote these notes with a lot of details, trying to help the reader as much as possible without losing clarity, but I don't deny the possibility that the difficulty would need a bit of tuning.

**Chapter 1**- This chapter contains the proof of a Theorem by Kunen and the definitions of I3 and I1. Version: 08-04-2011.
**Chapter 2**- This chapter contains an analysis (by D.A.Martin and R.Laver) of the space between I3 and I1. Version: 14-06-2011.
**Chapter 3**- This chapter contains the definition of I0 and an analysis of the properties of the elementary embeddings involved. Version: 20-09-2011.
**Chapter 4**- This chapter contains an analysis (by R.Laver) of the space between I1 and I0. Version: 31-07-2013.
**Chapter 5**- This chapter contains similarities between I0 and Determinacy in $L(\mathbb{R})$. Version: 05-01-2012.
**Chapter 6**- This chapter will contain an analysis of the Icarus sets.

Lecture notes for two PhD courses I gave at the Universities of Torino and Udine. The objective was to analyse the large cardinal consequences of the Axiom of Determinacy, up until the fact that the consistency of AD implies the consistency of an inaccessible limit of measurable cardinals. The only requirement for reading is a very basic knowledge of set theory. (Disclaimer: there are probably some errors since it is a first draft).

This are some notes by Liuzhen Wu and me about a slight modification of the Prikry condition, and its proofs in the case of some Prikry forcing. It can be seen also as an introduction to the Prikry condition for not-so-basic Prikry forcings, like the long extender one.