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.