Cohomology of Sheaves I. Homological Algebra 1. Exact categories 2. Homology of complexes 3. Additive categories 4. Homotopy theory of complexes 5. Abelian categories 6. Injective resolutions 7. Right derived functors 8. Composition products 9. Resume of the projective case 10. Complexes of free abelian groups 11. Sign rules II. Sheaf Theory 0. Direct limits of abelian groups 1. Presheaves and sheaves 2. Localization 3. Cohomology of sheaves 4. Direct and inverse image of sheaves. f* f* 5. Continuous maps and cohomology! 6. Locally closed subspaces h!h 7. Cup products 8. Tensor product of sheaves 9. Local cohomology 10. Cross products 11. Flat sheaves 12. Hom(E F) III. Cohomology with Compact Support 1. Locally compact spaces 2. Soft sheaves 3. Soft sheaves on $$\\mathbb {R}$$n 4. The exponential sequence 5. Cohomology of direct limits 6. Proper base change and proper homotopy 7. Locally closed subspaces 8. Cohomology of the n sphere 9. Dimension of locally compact spaces 10. Wilder’s finiteness theorem IV. Cohomology and Analysis 1. Homotopy invariance of sheaf cohomology 2. Locally compact spaces countable at infinity 3. Complex logarithms 4. Complex curve integrals. The monodromy theorem 5. The inhomogenous Cauchy Riemann equations 6. Existence theorems for analytic functions 7. De Rham theorem 8. Relative cohomology 9. Classification of locally constant sheaves V. Duality with Coefficient in a Field 1. Sheaves of linear forms 2. Verdier duality 3. Orientation of topological manifolds 4. Submanifolds of $$\\mathbb {R}$$n of codimension 1 5. Duality for a subspace 6. Alexander duality 7. Residue theorem for n 1 forms on $$\\mathbb {R}$$n VI. Poincare Duality with General Coefficients 1. Verdier duality 2. The dualizing complex D 3. Lefschetz duality 4. Algebraic duality 5. Universal coefficients 6. Alexander duality VII. Direct Image with Proper Support 1. The functor f! 2. The Künneth formula 3. Global form of Verdier duality 4. Covering spaces 5. Local form of Verdier duality VIII. Characteristic Classes 1. Local duality 2. Thom class 3. Oriented microbundles 4. Cohomology of real projective space 5. Stiefel Whitney classes 6. Chern classes 7. Pontrjagin classes IX. Borel Moore Homology 1. Proper homotopy invariance 2. Restriction maps 3. Cap products 4. Poincare duality 5. Cross products and the Künneth formula 6. Diagonal class of an oriented manifold 7. Gysin maps 8. Lefschetz fixed point formula 9. Wu’s formula 10. Preservation of numbers 11. Trace maps in homology X. Application to Algebraic Geometry 1. Dimension of algebraic varieties 2. The cohomology class of a subvariety 3. Homology class of a subvariety 4. Intersection theory 5. Algebraic families of cycles 6. Algebraic cycles and Chern classes XI. Derived Categories 1. Categories of fractions 2. The derived category D (A) 3. Triangles associated to an exact sequence 4. Yoneda extensions 5. Octahedra 6. Localization. This text exposes the basic features of cohomology of sheaves and its applications. The general theory of sheaves is very limited and no essential result is obtainable without turn­ ing to particular classes of topological spaces. The most satis­ factory general class is that of locally compact spaces and it is the study of such spaces which occupies the central part of this text. The fundamental concepts in the study of locally compact spaces is cohomology with compact support and a particular class of sheaves the so called soft sheaves. This class plays a double role as the basic vehicle for the internal theory and is the key to applications in analysis. The basic example of a soft sheaf is the sheaf of smooth functions on ~n or more generally on any smooth manifold. A rather large effort has been made to demon­ strate the relevance of sheaf theory in even the most elementary analysis. This process has been reversed in order to base the fundamental calculations in sheaf theory on elementary analysis.