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This text is an expanded version of the lecture notes of a minicourse (withthe same title of this text) delivered by the authors in the Bedlewo school"Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory"(from 4 to 16 July, 2011).  In the first part of this text, i.e., from Sections 1 to 5, we discuss theTeichm\"uller and moduli space of translation surfaces, the Teichm\"uller flowand the SL(2,R)-action on these moduli spaces and the Kontsevich-Zorich cocycleover the Teichm\"uller geodesic flow. We sketch two applications of the ergodicproperties of the Teichm\"uller flow and Kontsevich-Zorich cocycle, withrespect to Masur-Veech measures, to the unique ergodicity, deviation of ergodicaverages and weak mixing properties of typical interval exchangetransformations and translation flows. These applications are based on thefundamental fact that the Teichm\"uller flow and the Kontsevich-Zorich cocyclework as \emph{renormalization dynamics} for interval exchange transformationsand translation flows.  In the second part, i.e., from Sections 6 to 9, we start by pointing out thatit is interesting to study the ergodic properties of the Kontsevich-Zorichcocycle with respect to invariant measures other than the Masur-Veech ones, inview of potential applications to the investigation of billiards in rationalpolygons (for instance). We then study some examples of measures for which theergodic properties of the Kontsevich-Zorich cocycle are very different from thecase of Masur-Veech measures. Finally, we end these notes by constructing someexamples of closed SL(2,R)-orbits such that the restriction of theTeichm\"uller flow to them has arbitrary small rate of exponential mixing, or,equivalently, the naturally associated unitary SL(2,R)-representations havearbitrarily small spectral gap (and in particular it has complementary series).

Introduction to Teichm\"uller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards