Another cornerstone for gauge theories and gravity is the Lie group and Lie algebra. In this utility chapter, we attempt a comprehensive study of simple and compact Lie algebras and irreducible representations thereof. After introducing the roots and the weights, we make a half-hearted attempt at the classification, mainly to learn how to make use of the basic language. Simple roots, fundamental weights, Dynkin diagrams, and Dynkin labels are among the key concepts and tools involved. More mathematical facts immediately useful for gauge field theories are collected in the last section; various trace formulae, (dual-)Coxeter labels and numbers, and the notion of Dynkin indices are covered, all of which will play essential roles in later chapters.