Kobayashi metric and volume on varieties of general type

In my talk I will give the definition and basic properties of Kobayashi $p$-pseudometrics on exterior powers of the cotangent bundle of a complex manifold. Also I will show that positivity of these bundles implies hyperbolicity (theorems of Kobayashi, Kobayashi-Ochiai). Finally, I will sketch a proof of hyperbolicity of general high-degree hypersurfaces in the projective space (after Brotbek).