Category Theory -- Quotient categories -- Definition of KO(A) and some examples -- Krull-Schmidt theorems and applications -- Definition of G(R) and examples -- The connection between KO(R) and GO(R) -- Localization and relation between GO(R) and GO(RS) -- KO of graded rings -- Spec (R) and H(R) -- Picard group and the determinant -- Basic topological remarks -- Chain complexes and the nilpotence of of -- Serre's theorem -- Cancerllation theorems -- K1(A) -- K2(R) -- The exact sequence of Ki's -- Further results on K1 and K0 -- Relations between algebraic and topological K theory.