Abstract: Multiple zeta values (MZV) are the numbers defined by the convergent series of the form

$$\zeta(s_1,s_2,...,s_k)=\sum_{n_1>n_2>...>n_k>0}^\infty \{1/(n_1^{s_1} >... n_k^{s_k})\}$$

for $s_i$ positive integers. For these real numbers there are some beautiful relations, some of them due to Euler, like $\zeta(2,1) = \zeta(3)$ or $\zeta(2n) = q\pi^{2n}$ for $q$ a rational number. In this lecture I will present some of the most famous conjectures about MZV and its relations. I will show how we try to see the truthfulness of this conjecture by looking at them until a small degree bounded by the capacity of the actual computers.