I will introduce some continuum incidence problems from fractal geometry for which much recent progress has been made through Fourier analysis. These problems include projections of fractal sets, intersections of fractal sets with planes, and the dual (Kakeya) versions of such problems concerning the Hausdorff dimension of sets containing 'many' lines or curves. Some of these problems are set in Euclidean space, and others in the Heisenberg group, but most are connected in some way to local smoothing properties of wave equations. Local smoothing is closely related to the areas known as decoupling theory and Fourier restriction, and much recent progress comes from applying tools from these areas.  I will also discuss some related problems involving Fourier dimension.