For a prime $p$, a number field $k$ is called $p$-rational if the Galois group of the maximal pro-$p$ extension of $k$ unramified outside $p$ is a free pro-$p$ group.
In this talk, I will first introduce the fundamental concepts in the theory of Galois groups with restricted ramification, focusing on pro-$p$ extensions. Then, I will discuss the key results on $p$-rationality and its significance in number theory, highlighting its implications for current research and open problems in the field.