The study of Diophantine equations is one of the oldest problems in mathematics, with many questions that remain open to this day. However, at the start of the 20th century, mathematicians began to realize that tools from algebraic geometry could provide significant insights into the integer and rational solutions of polynomial equations, leading to the development of Diophantine geometry. In this seminar I will introduce the key ideas of this field, beginning with classical examples such as linear and quadratic Diophantine equations. In the second part of the talk, I will present the Chabauty-Coleman method, a modern technique for studying rational points on curves of genus at least 2, which combines many ideas from geometry, algebra and analysis.