The octonions are the largest of the four normed division algebras, alongside the real numbers, complex numbers, and quaternions. While often overlooked due to their nonassociativity, they play a central role in various areas of mathematics, including algebra, geometry, and topology. This paper explores the octonions and their connections to Clifford algebras, spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and exceptional Lie groups. It also touches on their applications in quantum logic, special relativity, and supersymmetry. The octonions were discovered by John T. Graves, who was influenced by Hamilton's work on quaternions. Graves introduced the concept of octaves, an 8-dimensional algebra that is a normed division algebra. This algebra was later recognized as the octonions, and its properties were further explored by Arthur Cayley. The octonions are nonassociative, a property that distinguishes them from the other normed division algebras. The octonions have significant implications in mathematics and physics. They are related to the exceptional Lie groups, which are important in the study of symmetry. The octonions also play a role in the classification of simple Jordan algebras and in the study of projective spaces. In topology, the octonions are connected to Bott periodicity, a fundamental concept in the study of homotopy groups. The octonions can be constructed in several ways, including through their multiplication table, the Fano plane, the Cayley-Dickson construction, and Clifford algebras. Each method provides a different perspective on the octonions and their properties. The Cayley-Dickson construction, in particular, shows how the octonions can be derived from the quaternions, and it highlights the nonassociativity of the octonions. The octonions are also important in the study of spinors and trialities. Spinors are mathematical objects that arise in the study of rotations and have applications in physics. The concept of triality, which is a more complex form of duality, is closely related to the octonions and their role in the classification of exceptional Lie groups. In summary, the octonions are a unique and important algebraic structure that has deep connections to various areas of mathematics and physics. Their nonassociativity and special properties make them a subject of ongoing research and exploration.The octonions are the largest of the four normed division algebras, alongside the real numbers, complex numbers, and quaternions. While often overlooked due to their nonassociativity, they play a central role in various areas of mathematics, including algebra, geometry, and topology. This paper explores the octonions and their connections to Clifford algebras, spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and exceptional Lie groups. It also touches on their applications in quantum logic, special relativity, and supersymmetry. The octonions were discovered by John T. Graves, who was influenced by Hamilton's work on quaternions. Graves introduced the concept of octaves, an 8-dimensional algebra that is a normed division algebra. This algebra was later recognized as the octonions, and its properties were further explored by Arthur Cayley. The octonions are nonassociative, a property that distinguishes them from the other normed division algebras. The octonions have significant implications in mathematics and physics. They are related to the exceptional Lie groups, which are important in the study of symmetry. The octonions also play a role in the classification of simple Jordan algebras and in the study of projective spaces. In topology, the octonions are connected to Bott periodicity, a fundamental concept in the study of homotopy groups. The octonions can be constructed in several ways, including through their multiplication table, the Fano plane, the Cayley-Dickson construction, and Clifford algebras. Each method provides a different perspective on the octonions and their properties. The Cayley-Dickson construction, in particular, shows how the octonions can be derived from the quaternions, and it highlights the nonassociativity of the octonions. The octonions are also important in the study of spinors and trialities. Spinors are mathematical objects that arise in the study of rotations and have applications in physics. The concept of triality, which is a more complex form of duality, is closely related to the octonions and their role in the classification of exceptional Lie groups. In summary, the octonions are a unique and important algebraic structure that has deep connections to various areas of mathematics and physics. Their nonassociativity and special properties make them a subject of ongoing research and exploration.