This lecture is an introduction to the analytic approach to the study of positivity of direct images. I will first explain Berndtsson's famous curvature formula of the direct images, and several important generalisations. In the second part, I will explain some applications in complex geometry: transcendental weak sub-adjunction formula, Iitaka conjecture etc. If the time is permitted, I will explain Berndtsson, Paun and Wang's curvature formula of higher direct images and its applications which is closely related to Ya Deng's previous lectures about the hyperbolicity of moduli spaces.