  These notes represent a summary of the introductory part of a course on
modified gravity delivered at several Spanish Universities (Granada, Valencia,
and Valladolid), at the University of Wisconsin-Milwaukee (WI, USA), and at the
Karl-Franzens Universitaet (Graz, Austria) during the period 2008-2011. We
begin with a discussion of the classical Newtonian framework and how special
relativity boosted the interest on new theories of gravity. Then we focus on
Nordstrom's scalar theories of gravity and their influence on Einstein's theory
of general relativity. We comment on the meaning of the Einstein equivalence
principle and its implications for the construction of alternative theories of
gravity. We present the cosmic speedup problem and how $f(R)$ theories can be
constrained attending to their weak-field behavior. We conclude by showing that
Palatini f(R) and f(R,Q) theories can be used to address different aspects of
quantum gravity phenomenology and singularity problems.
