We say that a regular semigroup $S$ is weakly generated by a set $X$ if it has no proper regular subsemigroup containing $X$. This does not mean that $S$ is generated by $X$. Very often, the subsemigroup generated by $X$ is a proper non-regular subsemigroup of $S$. In this talk we will begin by explaining our motivation for studying the structure of regular semigroups weakly generated by a set $X$. We will then focus only on the case of the regular semigroups weakly generated by idempotents. We will see there exists the "free" regular semigroup $FI(X)$ weakly generated by a set $X$ of idempotents, in the sense that all other regular semigroups weakly generated by the set $X$ of idempotents are homomorphic images of $FI(X)$. We will introduce $FI(X)$ by a presentation $\langle G,R\rangle$, where both $G$ and $R$ are infinite sets. Nevertheless, the word problem is decidable for this presentation as we can get a canonical form for each of its congruence classes. We will end this talk giving more details about the structure of $FI(X)$.