The notion of a meromorphic real normalized differential is equivalent to the notion of a harmonic function on an algebraic curve with "algebraic type" singularities at punctures. A pair of such differentials determines the harmonic map of the complement on the curve to the punctures on (in) the two-dimensional real plane, which can be regarded as a generalization of the amoeba map of a plane algebraic curve. The moduli spaces of algebraic curves with a pair of real normalized meromorphic differentials are fundamental in the theory of integrable systems and their perturbations, in the Seiberg-Witten solution to $N=2$ SUSY gauge models. Recently they found applications to the study of geometry of the moduli spaces of curves. In the talk I will present the basics of the theory of real normalized differentials and their applications and some open problems.