1 Mathematical Background -- 1.1 Computational Complexity -- 1.2 Probability -- 1.3 Markov Chains -- 1.4 Graph Theory -- 2 Techniques for Sampling and Approximate Sampling -- 2.1 Introduction -- 2.2 Direct Sampling -- 2.3 Markov Chain Method -- 3 Approximate Counting -- 3.1 Parsimonious Reductions -- 3.2 Counting Directly -- 3.3 Counting and Sampling -- 3.4 The Markov Chain Monte Carlo Method -- 4 Applications: Coupling -- 4.1 Hypergraph Colourings -- 4.2 Sink-Free Graph Orientations and Twice-Sat -- 4.3 Log-Concave Sampling, and the Volume of a Convex Body -- Intermezzo: Path Coupling -- 5 Applications: Path Coupling -- 5.1 Introduction -- 5.2 Twice-Sat Revisited -- 5.3 Sink- and Source-Free Graph Orientations -- 5.4 Totally Edge Cyclic Orientations -- 5.5 Independent Sets: The Conserved Hard-Core Model -- 5.6 Independent Sets: The Non-Conserved Hard-Core Model -- 5.7 Linear Extensions of a Partial Order -- 5.8 Graph Colouring -- 5.9 The Extended Potts Framework -- 5.10 Graph Colouring Revisited -- 6 Directions for Future Work -- 6.1 Breaking Thresholds -- 6.2 Beyond Self-Reducibility -- 6.3 Mixed Methods for Approximate Counting -- 6.4 Faster Reductions from Approximate Counting to Approximate Sampling -- 6.5 Anti-ferromagnetic Models -- 6.6 Log-Concave Sampling via Path Coupling -- Appendices -- A An Application of Dobrushin's Uniqueness Criterion -- B A Hierarchy of #SAT Restrictions -- B.1 Introduction -- B.2 A Summary of Known Results -- B.2.1 Easy Exact Counting -- B.2.2 Hard Exact Counting -- B.2.3 Easy Approximate Counting -- B.2.4 Hard Approximate Counting -- B.3 Summary and Conclusions -- C Equivalence of Transposition Distance to Spearman's Footrule.