Part I: Higgs bundles and the Non Abelian Hodge correspondence

Riemann surfaces and holomorphic bundles
Two classical theorems
Higher rank holomorphic bundles
Harmonic mappings
Non abelian Hodge correspondence
The Hitchin fibration and its fibers
The Hitchin section Theorem
Symmetric Spaces
Non positively curved manifolds
The heat flow method and harmonic mappings
Existence Theorem for the solutions of Hitchin Self Duality equations

Part II: Anosov representations and positivity

Background on hyperbolic surfaces
Contracting maps and Anosov bundles
Anosov representations: definition and limit maps
Orbit equivalence, reparametrisation and periods
The geodesic flow of an Anosov representation, with applications