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.

Let $V_n$ be the Segre embedding of ${\mathbb {P}^1}\times \cdots \times {\mathbb {P}^1}$ ( $n$ times). We prove that the higher secant varieties $\sigma _s(V_n)$ always have the expected dimension, except for $\sigma _3(V_4)$ , which is of dimension 1 less than expected.

In this article, we compute the dimension of all the higher secant varieties to the Segre–Veronese embedding of ℙ ⁿ × ℙ¹ via the section of the sheaf 𝒪(a, b) for any n, a, b ∈ ℤ⁺. We relate this result to the Grassmann Defectivity of Veronese varieties and we classify all the Grassmann (1, s − 1)-defective Veronese varieties.