ABSTRACT -- I. EQUIVARIANT SINGULAR THEORY -- 1. COEFFICIENT SYSTEMS -- 2. THE EXISTENCE THEOREMS FOR EQUIVARIANT SINGULAR HOMOLOGY AND C0H0M0L0GY -- 3. CONSTRUCTION OF EQUIVARIANT SINGULAR HOMOLOGY -- 4. CONSTRUCTION OF EQUIVARIANT SINGULAR COHOMOLOGY -- 5. THE H0M0T0PY AXIOM -- 6. THE EXCISION AXIOM -- 7. THE DIMENSION AXIOM -- 8. ADDITIVITY PROPERTIES -- II. FURTHER PROPERTIES OF EQUIVARIANT SINGULAR HOMOLOGY AND COHOMOLOGY -- 1. NATURALITY IN THE GROUP -- 2. TRANSFER HOMOMORPHISMS -- 3. THE KRONECKER INDEX AND THE CUP-PRODUCT -- BIBLIOGRAPHY