In this thesis we generalise quantum skew Howe duality to Lie superalgebras in type A, and show how this gives a categorification of certain representation categories of $\mathfrak{gl}(m|n)$. In particular, we use skew Howe duality to describe a category of representations generated monoidally by the exterior powers of the fundamental representation. This description is in terms of MOY diagrams, with one additional local relation on $n+1$ strands. This generalises the $n=0$ case from Cautis, Kamnitzer and Morrison. Using this, we give a categorification of this category in terms of foams, which generalises that of Queffelec, Rose and Lauda in the case $n=0$. The Reshetikhin-Turaev procedure gives a knot polynomial associated to $\mathfrak{gl}(m|n)$, which is a specialisation of the HOMFLY polynomial $P(a,q)$ at $a=q^{m-n}$. For the case $n=0$, the polynomial can be described nicely in terms of MOY diagrams, and therefore is related strongly to skew Howe duality. This was used by Queffelec and Rose to define $\mathfrak{sl}(n)$ Khovanov-Rozansky homology by categorified skew Howe duality. For general $n$, the relationship is less nice, and skew Howe duality is not sufficient to describe a homology theory associated with $\mathfrak{gl}(m|n)$ from our approach. Part of the problem is that the representation category no longer contains duals of the fundamental representations, which means that although a braid has an image in this categorified representation category, it is not possible to close this braid in the same way that Queffelec and Rose do. However, the categorified representation category does give partial progress towards the problem of defining a quantum categorification of the Alexander polynomial.