Let $(M_p)_{p \in \mathbb{N}}$ and $(A_p)_{p \in \mathbb{N}}$ be two sequences of positive real numbers. The space $\mathcal{S}^{\{M_p\}}_{\{A_p\}}$ consists of all $\varphi \in C^\infty(\mathbb{R})$ such that
$$\sup_{p,q \in \mathbb{N}} \sup_{x \in \mathbb{R}^d} \frac{|x^q\varphi^{(p)}(x)|}{h^{p+q}M_pA_q} < \infty\qquad {(*)}$$
for some $h > 0$. Similarly, the space $\mathcal{S}^{(M_p)}_{(A_p)}$ consists of all $\varphi \in C^\infty(\mathbb{R})$ such that $(*)$ holds for all $h > 0$. These spaces were introduced by Gelfand and Shilov in the 1950's and, in their honour, they are nowadays called Gelfand-Shilov spaces.  Gelfand and Shilov put forward the problem of characterizing the non-triviality of the spaces $\mathcal{S}^{\{M_p\}}_{\{A_p\}}$ and $\mathcal{S}^{(M_p)}_{(A_p)}$ in terms of the sequences $(M_p)_{p \in \mathbb{N}}$ and $(A_p)_{p \in \mathbb{N}}$. They themselves showed that $\mathcal{S}^{\{p!^\alpha\}}_{\{p!^\beta\}}$, $\alpha, \beta > 0$, is non-trivial if and only if $\alpha + \beta \geq 1$ but, in general, the problem still seems to be open. In the first part of this talk I will give an overview of several known partial solutions to this problem while, in the second part, I will report on recent work (joint with Jasson Vindas) in which we completely solved the problem for $M_p = p!$ fixed.