Classical Gaussian hypergeometric functions have been a rich vein of mathematics since their inception, with applications ranging as far as differential equations, spherical geometry, and modern number theory. In 1987, John Greene defined finite field analogs of these functions. In this talk, we will discuss these analogs as well as some of their connections to important arithmetic objects. After giving their construction, we will show how they can be used to count points on abelian varieties over finite fields. We will also discuss connections to the trace of Frobenius of certain Galois representations.