Abstract: Hitchin moduli spaces arise as the moduli space of vacua of certain N=2 4-dimensional supersymmetric field theories compactified on a circle. The Hitchin moduli space is actively studied in mathematics because of its particularly rich geometric structure. In this talk, I will focus on the asymptotic geometry of the natural hyperkahler metric on the Hitchin moduli space. Gaiotto-Moore-Neitzke proposed a construction for the hyperkahler metric using the count of BPS states in supersymmetric field theory. In particular, the natural hyperkahler metric is related to a simpler semiflat hyperkahler metric by quantum corrections given in terms of the BPS states. In this talk, I'll describe recent progress on proving Gaiotto-Moore-Neitzke's picture. I'll focus on two examples where the Hitchin moduli space has dimension four.