Abstract: In this talk, we will present an estimate on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations. This convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we obtained similar results for affine skew product toral transformations and, in the case of one dimensional torus translation, these estimates are nearly optimal. This is a joint work with Xiao-Chuan Liu and Silvius Klein.