In this work we derive limit theorems for trawl processes. First,we study the asymptotic behaviour of the partial sums of the discretized trawl process $(X_{iΔ_{n}})_{i=0}^{\lfloor nt\rfloor-1}$, under the assumption that as $n\uparrow\infty$, $Δ_{n}\downarrow0$ and $nΔ_{n}\rightarrowμ\in[0,+\infty]$. Second, we derive a functional limit theorem for trawl processes as the Lévy measure of the trawl seed grows to infinity and show that the limiting process has a Gaussian moving average representation.