Talagrand's influence inequality (1994) is an asymptotic improvement of the classical L₂ Poincaré inequality on the Hamming cube {-1,1}ⁿ with numerous applications to Boolean analysis, discrete probability theory and geometric functional analysis. In these talks, we shall discuss various refinements of Talagrand's inequality, including its L_p analogues and Banach space-valued versions. Emphasis will be given to the probabilistic aspects of the proofs. We will also explain a geometric application of these new refinements to the bi-Lipschitz embeddability of a natural family of finite metrics and mention related open problems.