  Traditional computers work with finite numbers. Situations where the usage of
infinite or infinitesimal quantities is required are studied mainly
theoretically. In this paper, a recently introduced computational methodology
(that is not related to the non-standard analysis) is used to work with finite,
infinite, and infinitesimal numbers \textit{numerically}. This can be done on a
new kind of a computer - the Infinity Computer - able to work with all these
types of numbers. The new computational tools both give possibilities to
execute computations of a new type and open new horizons for creating new
mathematical models where a computational usage of infinite and/or
infinitesimal numbers can be useful. A number of numerical examples showing the
potential of the new approach and dealing with divergent series, limits,
probability theory, linear algebra, and calculation of volumes of objects
consisting of parts of different dimensions are given.
