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.