The periodization of functions is well known in harmonic analysis (in particular, in application to sampling of signals), its central point being the Poisson summation formula.The notion of periodization for fractional integration appears in the paper H. Weyl (1917) well known to experts in Fractional Calculus. The periodic fractional integral (Weyl integral) of a 2π-periodic function f(x), generally speaking, coincides with the properly interpreted Liouville fractional integral of f. In fact, this is nothing else but the statement that the periodic Weyl fractional kernel is the periodization of the Liouville kernel.We present a general approach to the periodization of functions of one variable, but the main focus is made on the periodization of fractional integrals of functions of two variables. There exist many forms of multidimensional fractional integro-differentiation. We dwell on the case of the Riesz fractional integrals (Riesz potentials) of two variables and show that already in this case we come across different effects, depending on whether we use the repeated periodization, first in one variable, and afterwards in another one, or the so called double periodization. We show that the naturally introduced doubly-periodic Weyl-Riesz kernel of order 0 <α 2 in general coincides with the periodization of the Riesz kernel, the repeated periodization being possible for all 0 <α < 2, while the double one is applicable only for 0 <α 1. This is obtained as a realization of a certain general scheme of periodization (repeated or double periodization).