Let X be a smooth projective variety, x_1, …, x_n general points, \beta a curve class on X and (C,p_1,…,p_n) a general smooth pointed curve. We consider the following problem: how many maps f:C\to X in class \beta are there such that f(p_i)=x_i? (geometric Tevelev degrees). When the problem is instead formulated in Gromov-Witten theory (virtual Tevelev degrees), the answers seem to be simpler, accessible in terms of the quantum cohomology of X, but not always enumerative. I will discuss two lines of inquiry, both of which are ongoing: (1) when X and \beta are sufficiently positive, the virtual and geometric counts agree (joint with Pandharipande), and (2) the computation of geometric Tevelev degrees for X=P^r, via limit linear series, Schubert calculus, and complete collineations (joint with Farkas).