Cryptographic applications of multilinear maps beyond bilinear pairings were first proposed in the work of Boneh and Silverberg. However the existence of cryptographically interesting $n$-multilinear maps for $n > 2$ remains an open problem. Very recently Lin and Tessaro showed that trilinear maps are sufficient for the purpose of achieving indistinguishability obfuscation. This striking result put spotlight on the following question: can a cryptographically interesting algebraic trilinear map be constructed? In this talk we discuss a method for constructing such a trilinear map, and present concrete candidate trilinear maps which involve Weil descent and the Jacobian varieties of hyperelliptic curves. A preprint is available here: https://arxiv.org/abs/1810.03646.