This report presents and discusses proofs of the Malgrange-Ehrenpreis theorem, which states that every non-zero linear partial differential operator with constant coefficients has a fundamental solution. The main topic is explicit formulae, and more specifically, how they can be used to prove the theorem. Two different formulas will be considered in detail and the aim is to provide a fundamental and elementary description of how to prove the Malgrange-Ehrenpreis theorem using those formulas. In addition to the proofs, an example of how to use one of the formulas for the Cauchy-Riemann operator is shown. Finally, the report also contains a chapter discussing a few different notable methods of proof and their historical signifance.