In the first part of this seminar, I'll give a brief review of the fundamental role played by "dualities" in different physical contexts. Then I'll focus on a particular duality, peculiar of string theory, which hints that string physics at a very small length scale cannot be distinguished from physics at a large one: this is the so-called "target space duality", meaning by "target space" simply the space-time in which the string moves. An attempt to make such a symmetry explicitly manifest in the action of the bosonic string theory leads to a "doubling" of the string coordinates, with the ordinary space-time string coordinates plus their "duals" treated on the same footing. It results that they behave as "non-commuting" phase-space type coordinates. The interest in such a formulation is especially in exploring how the effective gravitational string action becomes in the light of such doubling of coordinates, hopefully shedding light on aspects of string gravity unexplored thus far.