The density of a permutation pattern $\pi$ in a permutation $\sigma$ is the proportion of subsequences of $\sigma$ of length $|\pi|$ that are isomorphic to $\pi$. The maximal value of the density is found for several patterns $\pi$, and asymptotic upper and lower bounds for the maximal density are found in several other cases. The results are generalised to sets of patterns and the maximum density is found for all sets of length $3$ patterns.