What are the most fundamental/useful/interesting ways in which the concepts of Brownian motion, martingales and markov chains are related?
I'm a graduate student doing a crash course in probability and stochastic analysis. At the moment, the world of probability is a confusing blur, but I'm starting with a grounding in the basic theory of markov chains, martingales and Brownian motion. While I've done a fair amount of analysis, I have almost no experience in these other matters and while understanding the definitions on their own isn't too difficult, the big picture is a long way away.
I would like to gather together results and heuristics, each of which links together two or more of Brownian motion, martingales and Markov chains in some way. Answers which relate probability to real or complex analysis would also be welcome, such as "Result X about martingales is much like the basic fact Y about sequences".
The thread may go on to contain a Big List in which each answer is the posters' favourite as yet unspecified result of the form "This expression related to a markov chain is always a martingale because blah. It represents the intuitive idea that blah".
Because I know little, I can't gauge the worthiness of this question very well so apologies in advance if it is deemed untenable by the MO police.