We investigate a variant of an old problem in linear algebra and operator theory that was popularized by Paul Halmos: Must almost commuting matrices be nearly commuting? To be more precise, we say that a pair of n-by-n complex valued matrices (A,B) are “almost commuting" if AB - BA is small in some sense. In the same manner, we say that a pair of n-by-n complex valued matrices (X,Y) are “nearly commuting" if X-A and Y-B are small in some sense and AB = BA. Although we will be exploring deep ideas in operator theory, only a basic understanding of undergraduate linear algebra and real analysis will be assumed. We will briefly discuss history of the problem, discuss the progress on the problem, and sketch the proof of a quantitative result which establishes that “almost commuting" matrices are “nearly commuting" for different types of matrices.