This paper presents the beginnings and historical development of ancient Greek mathematics. The ancient Greeks influenced the development of science, philosophy and architecture, and their influence in the field of mathematics is still present nowadays. In the introductory part of the paper, the number systems of the ancient Greeks and their disadvantages are presented. Then the importance of the first Greek mathematician, Thales of Miletus, who used his theorems to solve many practical problems of that time, is described. Thales' student Pythagoras and his students reached significant results in the field of number theory and geometry, proved Pythagoras' teaching and discovered irrational numbers. The school of Pythagoras influenced Plato, who founded his own school, the famous Academy. Arithmetic, trigonometry and planimetry were studied at the Academy. One of the most famous students of the Academy, Aristotle, made the greatest contribution in the field of logic. A special emphasis is placed on the contribution of the most renowned Greek mathematician Euclid, known as the founder of geometry. Euclid left 13 books incorporated into one great work called Elements. For centuries, Elements was a basic mathematics textbook and one of the most studied works. Some of the definitions, postulates, axioms and propositions presented in the Elements are listed, and the topics in each of the 13 books are briefly described. After Euclid, the contribution of Archimedes is described, who marks the end of the ancient Greek age of mathematics. The paper ends with a presentation of three classical problems of ancient Greek mathematics.