Moduli problems are of central importance in algebraic geometry. Among them, the moduli spaces of stable maps are foundations for Gromov-Witten theory. In this talk, I will focus on the deep geometry of the moduli spaces of stable maps. These moduli spaces are arbitrarily singular. The resolution of singularity is arguably one of the hardest problems in algebraic geometry. To date, the problem remains open for positive characteristics. When the genus g=0, the stable map moduli are smooth. For g=1, the moduli spaces are singular and a resolution was constructed by Vakil and Zinger, followed by an algebraic approach of Hu and (Jun) Li. The method of Hu and Li was further developed jointly with Jingchen Niu to finally give a resolution in the case when g=2. Jointly obtained with Niu, I will further introduce the moduli of curves with twisted fields. These moduli provide natural geometric meaning for the resolution of Vakil-Zinger for genus 1 and for the resolution of Hu-Li-Niu for genus 2.