The computational complexity of matrix multiplication
Classical work of Coppersmith shows that for some $\alpha &gt; 0$, one can multiply an $n \times n^\alpha$ matrix with an $n^\alpha \times n$ matrix in $\tilde{O}(n^2)$ arithmetic operations. This is a crucial ingredient of Ryan Williams's recent celebrated result.

François le Gall recently improved on Coppersmith's work, and his paper has just been accepted to FOCS 2012. In order to understand this work, you will need some knowledge of algebraic complexity theory. Virginia Williams's paper contains some relevant pointers. In particular, Coppersmith's work is completely described in Algebraic Complexity Theory, the book.

A different strand of work concentrates on multiplying matrices approximately. You can check this work by Magen and Zouzias. This is useful for handling really large matrices, say multiplying an $n \times N$ matrix and an $N \times n$ matrix, where $N \gg n$.

The basic approach is to sample the matrices (this corresponds to a randomized dimensionality reduction), and multiplying the much smaller sampled matrices. The trick is to find out when and in what sense this gives a good approximation. In contrast to the previous strand of work which is completely impractical, sampling algorithms are practical and even necessary for handling large amounts of data.