My research is focused on functional inequalities related to the concentration of measure phenomenon, in particular the transport-entropy inequalities, also called transport inequalities. The chapters of this document are articulated around this inequality and its relation with other functional inequalities. We present an abstract general version of the measure concentration principle in order to unify the different results of my research. Moreover some elementary arguments are given for a better understanding of the mathematical tools. The first chapter introduces the concentration of measure principle. It's general setting allows to compare many results depending on the involved cost function. The second chapter is devoted to the so-called weak transport inequality related to the general concentration of measure principle. We also present the links between logarithmic Sobolev inequalities and transport inequalities in metric spaces. The third chapter develops different results around the so-called barycentric costs, the notion of convex order on probability measures and weak-transport inequalities associated to this barycentric cost. Chapter IV is based on examples of ``universal'' transport inequalities as the so-called Csizár-Kullback-Pinsker inequality and, by tensorization, their applications to Bernstein type of deviation inequalities for suprema of empirical processes. We also present recent transport inequalities obtained for the uniform law on the symmetric group obtained by using different tensorization arguments. Chapter V is about the Poincaré inequality and its characterization in terms of a weak dimension-free concentration principles. Chapter VI concerns the notion of curvature on metric spaces and functional inequalities as the so-called displacement convexity property. We explain how a concentration property may imply a logarithmic Sobolev inequality or a Poincaré inequality when this convexity property holds on a geodesic space. In another direction, we try to extend the notion of curvature to discrete spaces by considering a similar convexity property of entropy on discrete spaces. The last chapter is devoted to improved versions of the logarithmic Sobolev inequality and the HWI inequality for the standard Gaussian measure on $\R^n$.