This lecture shall be addressed to mathematicians interested in Number Theory in general. We shall discuss first the asymptotic behavior of heights with respect to rational transformations, when the involved heights are large, explaining very briefly how this can be applied to problems of rational and integral points on algebraic curves.

Then we shall discuss points of small height, recalling some lower bounds, and also how a very small height implies certain Galois equidistributions. We shall also discuss the relation of this with classical conjectures (now solved) by Lang, Manin, Mumford.