This master's thesis is focussed around investigating Massey products as tools for studying properties of links, in particular the Brunnian property. In the literature, there are only a few examples of the Massey product being used to study linking, none of which has any emphasis on links with the Brunnian property, except for computations for the Borromean rings. The result of the work is a number of thorough computations of Massey products in link complements, with the negative conclusion that the Massey product does not detect the Brunnian property.