The distribution of prime numbers has been one of the most fascinating and challenging problems in number theory for centuries. This paper examines the behavior of prime numbers within short intervals, exploring both classical results and modern developments in the field. We investigate the prime number theorem, the Riemann hypothesis, and various conjectures related to prime gaps and their distribution patterns. Through analytical methods and computational evidence, we explore how primes are distributed in intervals of the form [x, x + h] where h is small relative to x. The study reveals significant insights into the irregularity and randomness of prime distribution while highlighting the deep connections between prime number theory and other areas of mathematics.