A Besicovich set is a compact subset of R^d that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that every Besicovich set in R^d must have dimension d. I will discuss some new trilinear Kakeya-type bounds in R⁴, and how these bounds can be used to obtain improved bounds on the dimension of certain sets in R⁴ that resemble Kakeya sets. This is joint work with Larry Guth.