Dive into the world of non-commutative geometry with this comprehensive lecture, the first in a three-part series delivered by John Barrett at the Galileo Galilei Institute. Over the course of 1 hour and 41 minutes, explore fundamental concepts such as algebra, reconstruction theorems, and differential operators. Gain insights into fuzzy spaces, Hermitian operators, and representations. Examine linear terms, Casimir operators, and eigenvectors. The lecture concludes with a Q&A session, allowing for deeper understanding and clarification of complex topics. Perfect for mathematics enthusiasts and students seeking to expand their knowledge in advanced geometric theories.