This is the expanded second edition of a successful textbook that provides a broad introduction to the important area of stochastic modelling. The original text had been developed from lecture notes for a one-semester course on the topic for third-year science and actuarial students at the University of Melbourne. It reviews the basics of probability theory, and then covers the following topics: Markov chains, Markov decision processes, jump Markov processes, elements of queueing theory, basic renewal theory, elements of time series and simulation. The present edition adds new chapters on elements of stochastic calculus and introductory mathematical finance that logically complement the topics chosen for the first edition. This makes the book suitable for a larger variety of university courses presenting the fundamentals of modern stochastic modelling. Rigorous proofs are often replaced with sketches of arguments - with indications as to why a particular result holds, and also how it is connected to other results - and illustrated by well-selected examples.Wherever possible, the book includes references to more specialised texts containing both proofs and more advanced material related to the topics covered.