  We introduce and study properties of the Terracini locus of projective
varieties X, which is the locus of finite subsets S of X such that 2S fails to
impose independent conditions to a linear system L. Terracini loci are relevant
in the study of interpolation problems over double points in special position,
but they also enter naturally in the study of special loci contained in secant
varieties to projective varieties. We find some criteria which exclude that a
set S belongs to the Terracini locus. Furthermore, in the case where X is a
Veronese variety, we bound the dimension of the Terracini locus and we
determine examples in which the locus has codimension 1 in the symmetric
product of X.
