A finite-field Kakeya set is a subset of (F_q)ⁿ (the n-dimensional vector space over a finite field of order q) which contains a line in each direction. I will first show how to use polynomials to obtain bounds on the size of a Kakeya set in finite vector spaces. Then I will show how to modify this polynomial technique to obtain better bounds on the size of these Kakeya sets. Time permitting, I will discuss bounds on related incidence problems.