This thesis is a study of classical and quantum gravity with Ashtekar variables. The Ashtekar constraints are shown to capture the essence of the constraints and constraint algebra of General Relativity in four dimensions. A classification scheme of the solution space of the Ashtekar constraints is proposed and the corresponding physics is investigated. The manifestly covariant equations of motion for the Ashtekar variables are derived. Explicit examples are discussed and new classical solutions of General Relativity are constructed by exploiting the properties of the Ashtekar variables.

Non-perturbative canonical quantization of the theory is performed. The ordering of the quantum constraints as well as the formal closure of the quantum constraint algebra are explored. A detailed Becchi-Rouet-Stora-Tyutin (BRST) analysis of the theory is given. The results demonstrate explicitly that in quantum gravity, fluctuations in topology can occur and there are strong evidences of phases in the theory. There is a phase which is described by a topological quantum field theory (TQFT) of the Donaldson-Witten type and an Abelian antiinstanton phase wherein self-interactions of the gravitational fields produce symmetry breaking from SO(3) to U(1). The full theory is much richer and includes fluctuations which bring the system out of the various restricted sectors while preserving diffeomorphism invariance. Invariants of the quantum theory with are constructed through BRST descents. They provide a clear and systematic characterization of non-local observables in quantum gravity, and can yield further differential invariants of four-manifolds.