Measure Theory
Theoretical foundations of measure theoretic probability, random variables, etc.

Convergence
Types of convergence and their relations

Laws of Large Numbers
Chebyshev's Theorem, weak laws, Borel-Cantelli lemmas, methods of proving almost sure convergence, convergence of random series, strong laws, laws under minimal hypotheses

Central Limit Theorems
Central limit theorem (DeMoivre-Laplace Theorem), Stirling's formula, weak convergence, Poisson convergence, characteristic functions

Discrete Probability and Combinatorics
Counting techniques, multiplication principle (Fundamental Principle of Counting), permutations, combinations, distinguishable and indistinguishable objects, with and without replacement, urn models, basic probability distributions, conditional probability, generating functions

Stochastic Processes
random walks, Markov chains, queuing processes, birth and death processes, branching processes

Recommended Texts:

An Introduction to Probability Theory and its Applications, volume I, 3^rd edition, by William Feller
Probability: Theory and Examples, 2^nd edition, by Richard Durrett
Probability and Measure, 3^rd edition, by Patrick Billingsley
A First Course in Stochastic Models, by Henk Tijms
Elements of Applied Stochastic Processes, by U.N. Bhat and G.K. Miller