This paper aims to develop a higher K-theory for additive categories with exact sequences, extending the Grothendieck group in a natural way. The paper introduces a category Q(M) derived from an additive category M embedded in an abelian category A, where morphisms are defined via isomorphisms of subquotients. The classifying space BQ(M) is then used to define K-groups via homotopy groups. The paper establishes that the fundamental group of BQ(M) is isomorphic to the Grothendieck group of M, motivating the definition of K-groups as homotopy groups of BQ(M). The first part of the paper develops the general theory of these K-groups, including tools for working with classifying spaces and results on exact sequences. It also defines K-groups for exact categories and proves that K₀(M) is isomorphic to the Grothendieck group of M. The paper then presents four key theorems—exactness, resolution, devissage, and localization—which generalize results for the Grothendieck group. The second part applies these results to rings and schemes. For a ring A, K₁(A) is defined as the K-groups of finitely generated projective A-modules, and it is shown that K₁(A) is isomorphic to K₁'(A) for regular rings. The paper also proves formulas for K'-groups of polynomial and Laurent polynomial rings over noetherian rings. For schemes, K-groups are defined using vector bundles or coherent sheaves, and a spectral sequence is introduced to compute K'-groups. The paper proves Bloch's formula for regular schemes of finite type over a field, showing that the Chow group of codimension p cycles is isomorphic to a cohomology group. It also computes K-groups of projective bundles and Severi-Brauer schemes. The paper includes proofs of all results announced in [Quillen 1], except for a theorem on H-spaces, which is used in the second part. The paper concludes with a discussion of the potential of higher K-theory to provide a theory of the Chow ring for non-quasi-projective regular varieties.This paper aims to develop a higher K-theory for additive categories with exact sequences, extending the Grothendieck group in a natural way. The paper introduces a category Q(M) derived from an additive category M embedded in an abelian category A, where morphisms are defined via isomorphisms of subquotients. The classifying space BQ(M) is then used to define K-groups via homotopy groups. The paper establishes that the fundamental group of BQ(M) is isomorphic to the Grothendieck group of M, motivating the definition of K-groups as homotopy groups of BQ(M). The first part of the paper develops the general theory of these K-groups, including tools for working with classifying spaces and results on exact sequences. It also defines K-groups for exact categories and proves that K₀(M) is isomorphic to the Grothendieck group of M. The paper then presents four key theorems—exactness, resolution, devissage, and localization—which generalize results for the Grothendieck group. The second part applies these results to rings and schemes. For a ring A, K₁(A) is defined as the K-groups of finitely generated projective A-modules, and it is shown that K₁(A) is isomorphic to K₁'(A) for regular rings. The paper also proves formulas for K'-groups of polynomial and Laurent polynomial rings over noetherian rings. For schemes, K-groups are defined using vector bundles or coherent sheaves, and a spectral sequence is introduced to compute K'-groups. The paper proves Bloch's formula for regular schemes of finite type over a field, showing that the Chow group of codimension p cycles is isomorphic to a cohomology group. It also computes K-groups of projective bundles and Severi-Brauer schemes. The paper includes proofs of all results announced in [Quillen 1], except for a theorem on H-spaces, which is used in the second part. The paper concludes with a discussion of the potential of higher K-theory to provide a theory of the Chow ring for non-quasi-projective regular varieties.