We start by introducing Brauer-Manin obstructions to local-global principles over varieties. We then move to some of the known literature on Brauer-Manin obstructions for Kummer surfaces of products of elliptic curves. We finally present our work on some of the special cases where we calculate the Brauer group of a Kummer surface $X=Kum(E \times E')$ of a product of CM elliptic curves $E$ and $E'$, where $End(E)=End(E')=\mathbb{Z}[\zeta_3]$, and show that a non-trivial 5-torsion element of the transcendental Brauer group gives rise to Brauer Manin obstruction to weak approximation for $X$.