Summary: Here is a set lectures by Sid Redner.
There exists several simple models of random walks. It can be either fixed stepsize but random directions or varying stepsize.
Fixed Stepsize A simple model is that we assume each step is a step of size $ \epsilon$, but with random directions. The position of this random walker at step $ N$ is the summation of all the steps (vectors),
$$ \vec X = \sum_i^N x_i, $$ where $ x_i$ is the vector that represents step i.