Let F_q be a finite field with q elements, and let G=G(F_q) be the group of F_q-points of a connected reductive group with connected center defined over F_q. The main topic of this talk is to address the following problem: Given an irreducible complex character chi of G, how does one tell from the parameters of chi whether chi is real-valued? I will begin by giving some motivation for this problem, and then give a description of the Jordan decomposition (or Lusztig parameterization) of the irreducible characters of G. In particular, this parameterization consists of a pair (s, psi), where s is a semisimple element of the dual group G*, and psi is a unipotent character of the centralizer of s in G*. I will then motivate the following conjecture on the classification of real-valued characters of G: the character corresponding to the pair (s, psi) is real-valued if and only if s is a real element, and if h is an element of G* which conjugates s to s^{-1}, then h also conjugates \psi to its complex conjugate. I will then outline a proof of this conjecture in the case that the centralizer of s in G* is a Levi subgroup. This material is ongoing joint work with Bhama Srinivasan from University of Illinois at Chicago.