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Traditional computers work numerically only with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this lecture, a new computational methodology (that is not related to non-standard analysis approaches) is described. It is based on the principle 'The part is less than the whole' applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The new methodology has allowed the author to introduce the Infinity Computer working numerically with infinite and infinitesimal numbers. The new computational paradigm both gives possibilities to execute computations of a new type and simplifies fields of Mathematics and Computer Science where infinity and/or infinitesimals are required. Examples of the usage of the introduced computational tools are given during the lecture.