  We consider here the problem, which is quite classical in Algebraic geometry,
of studying the secant varieties of a projective variety $X$. The case we
concentrate on is when $X$ is a Veronese variety, a Grassmannian or a Segre
variety. Not only these varieties are among the ones that have been most
classically studied, but a strong motivation in taking them into consideration
is the fact that they parameterize, respectively, symmetric, skew-symmetric and
general tensors, which are decomposable, and their secant varieties give a
stratification of tensors via tensor rank. We collect here most of the known
results and the open problems on this fascinating subject.
