In this talk, I will survey some aspects relating classes of PDEs with metrics on a 2-dimensional manifold with non zero constant Gaussian curvature. The notion of a differential equation (or system of equations) describing pseudo-spherical surfaces (curvature -1) or spherical surfaces (curvature 1) will be introduced. Such equations have remarkable properties. Each equation is the integrability condition of a linear problem explicitly given. The linear problem may provide solutions for the equation by using Bäcklund type transformations or by applying the inverse scattering method. Moreover, the geometric properties of the surfaces may provide infinitely many conservation laws. Very well known equations such as the sine-Gordon, Korteveg de Vries, Non Linear Schrödinger, Camassa-Holm, short-pulse equation, elliptic sine-Gordon, etc. are examples of large classes of equations related to metrics with non zero constant curvature. Classical and more recent results characterizing and classifying certain types of equations will be mentioned. Examples and illustrations will be included. Some higher dimensions generalizations will be mentioned.