The paper discusses Vassiliev invariants, also known as finite-type knot invariants, which are powerful tools in knot theory. These invariants are defined using the concept of weight systems, which are functions on chord diagrams. The paper shows that Vassiliev invariants are at least as powerful as the Jones polynomial and its generalizations from quantum groups. It also explores the relationship between Vassiliev invariants and Lie algebras, showing that many weight systems can be constructed from Lie algebraic information. The paper introduces the algebra A of diagrams, which is a graded vector space, and its primitive elements. It also discusses the co-algebra structure of A and its relation to weight systems. The paper concludes with conjectures about the classification of weight systems and their relationship to Lie algebras. The paper is structured into sections that cover the introduction, basic constructions, the algebra A of diagrams, Kontsevich's construction, the primitive elements of A, the size of A, W, and P, odds and ends, and a summary of spaces and maps. The paper also includes proofs of several theorems and lemmas, as well as exercises and remarks.The paper discusses Vassiliev invariants, also known as finite-type knot invariants, which are powerful tools in knot theory. These invariants are defined using the concept of weight systems, which are functions on chord diagrams. The paper shows that Vassiliev invariants are at least as powerful as the Jones polynomial and its generalizations from quantum groups. It also explores the relationship between Vassiliev invariants and Lie algebras, showing that many weight systems can be constructed from Lie algebraic information. The paper introduces the algebra A of diagrams, which is a graded vector space, and its primitive elements. It also discusses the co-algebra structure of A and its relation to weight systems. The paper concludes with conjectures about the classification of weight systems and their relationship to Lie algebras. The paper is structured into sections that cover the introduction, basic constructions, the algebra A of diagrams, Kontsevich's construction, the primitive elements of A, the size of A, W, and P, odds and ends, and a summary of spaces and maps. The paper also includes proofs of several theorems and lemmas, as well as exercises and remarks.