This paper is an essay on how the development of group theory led to the discovery of various families of simple groups, and how these in turn led to the theory of buildings. In outline the story is this. Galois first used the term 'group' in the technical sense, and found the first simple groups. Jordan, in his famous Traité des substitutions et des équations algébriques Ⓣ, published in 1870, promoted Galois' work and put the theory of groups on a firm foundation. At this time groups were treated as groups of permutations, but other aspects of group theory were soon on the way. Lie visited Paris in 1870 as a graduate student, and went on to create the theory of continuous transformation groups. Killing came to such groups independently, and in 1888 found the classification of the simple Lie groups, using semisimple complex Lie algebras (families A through G). Cartan refined this classification in 1894, correcting some errors in the proofs, and it is now known as the Killing-Cartan classification. The classical families (A through D) soon led to groups over fields other than the real or complex numbers, and a comprehensive study was published by Dickson in 1901. Later he dealt with $E_{6}$ and G, but progress on the others did not occur until after the Second World War. Tits was working on the problem, as was Chevalley, who was a more established mathematician at that time. Chevalley succeeded in 1955, and his paper was soon followed by variations due to Steinberg, Tits, Suzuki, and Ree. During this time Tits was gradually developing the theory of buildings, and his book "Buildings of spherical type and finite BN-pairs" in 1974 produced a fully-fledged theory that has since found many uses. ... we mention some of Tits' early work on buildings, and we discuss the contents of his above-mentioned book concerning buildings of spherical type. Finally ... a later approach to buildings, also due to Tits, is mentioned, and we return at the end to the construction of the exceptional groups of Lie type using building theory.

... for their profound achievements in algebra and in particular for shaping modern group theory.

Tits created a new and highly influential vision of groups as geometric objects. He introduced what is now known as a Tits building, which encodes in geometric terms the algebraic structure of linear groups. The theory of buildings is a central unifying principle with an amazing range of applications, for example to the classification of algebraic and Lie groups as well as finite simple groups, to Kac-Moody groups (used by theoretical physicists), to combinatorial geometry (used in computer science), and to the study of rigidity phenomena in negatively curved spaces. Tits's geometric approach was essential in the study and realisation of the sporadic groups, including the Monster. He also established the celebrated "Tits alternative": every finitely generated linear group is either virtually solvable or contains a copy of the free group on two generators. This result has inspired numerous variations and applications. The achievements of John Thompson and of Jacques Tits are of extraordinary depth and influence. They complement each other and together form the backbone of modern group theory.