Exponential sums were first introduced 200 years ago by Gauss. Since then, bounds on such sums have been an interesting item of study and have found several applications. We will give an introduction to such sums and then consider multilinear exponential sums, with the hope of giving some new upper bounds. The method of providing upper bounds will rely on some new bounds in additive combinatorics. We will therefore give a background of some recent ideas and some longstanding problems in additive combinatorics and show how these results can lead to new bounds on exponential sums. Explicitly, we will consider how new results on the number of collinear triples gives us new bounds on trilinear exponential sums. If there is time, we will also consider higher dimensional multilinear exponential sums and/or some applications to multinomial exponential sums.