Part I. The Type Problem: 1. Basic facts; 2. Recurrence and transience of infinite networks; 3. Applications to random walks; 4. Isoperimetric inequalities; 5. Transient subtrees, and the classification of the recurrent quasi transitive graphs; 6. More on recurrence; Part II. The Spectral Radius: 7. Superharmonic functions and r-recurrence; 8. The spectral radius; 9. Computing the Green function; 10. Spectral radius and strong isoperimetric inequality; 11. A lower bound for simple random walk; 12. Spectral radius and amenability; Part III. The Asymptotic Behaviour of Transition Probabilities: 13. The local central limit theorem on the grid; 14. Growth, isoperimetric inequalities, and the asymptotic type of random walk; 15. The asymptotic type of random walk on amenable groups; 16. Simple random walk on the Sierpinski graphs; 17. Local limit theorems on free products; 18. Intermezzo; 19. Free groups and homogenous trees; Part IV. An Introduction to Topological Boundary Theory: 20. Probabilistic approach to the Dirichlet problem, and a class of compactifications; 21. Ends of graphs and the Dirichlet problem; 22. Hyperbolic groups and graphs; 23. The Dirichlet problem for circle packing graphs; 24. The construction of the Martin boundary; 25. Generalized lattices, Abelian and nilpotent groups, and graphs with polynomial growth; 27. The Martin boundary of hyperbolic graphs; 28. Cartesian products.