What mathematical background is needed to study random walks rigorously? I've been studying random walks as part of a finance course and I'm starting to become really interested, and I'd like to start a study of them from a mathematics (i.e. more formal/rigorous) point of view. 
However, my knowledge of mathematics is quite limited (calculus, basic linear algebra and probability, a bit of (stochastic) differential equations). So I was wondering what fields I need to familiarise myself with before I can effectively study random walks. Maybe there's even a good self-contained book on random walks for beginners?
 A: You seem to know most of what is required.  I would get comfortable with the Central Limit Theorem, if you aren't already.
Here is a source on Random Walks you might start with: http://www.mit.edu/~kardar/teaching/projects/chemotaxis%28AndreaSchmidt%29/random.htm
A: I'm a little less optimistic than the other respondents as regards the required background to study mathematical finance.  I've seen a lot of students struggle in upper level courses due to weak backgrounds in analysis.
Of course, if you just want to stick to discrete time models, then you can get away with less background.  I'd recommend Shreve's Stochastic Calculus for Finance I: The Binomial Asset Pricing Model if you are going to take this route.
But if you serious about doing continuous time finance, you need to understand Brownian motion, quadratic variation, sigma algebras, etc. which means a good understanding of real analysis up through Lebesgue integration, and preferably a measure-theoretic introduction to probability.  If you try to do SDE's without that, it is just formal manipulation of expressions, without really understanding what is going on mathematically.
That said, a deep mathematical understanding is not per se necessary to succeed in the financial industry, so it depends on what your goals are.
