We study the Continuum Directed Random Polymer (CDRP) which arises as a universal scaling limit of discrete directed polymers. In this talk, I will present some of the recent progress in understanding the geometry of the CDRP. In particular, I will show CDRP exhibits pointwise localization and pathwise tightness under 2/3 scaling, confirming the existing predictions in polymer literature under continuous setting. I will explain how our results also shed light on certain properties of the KPZ equation (free energy of the CDRP) such as ergodicity and limiting Bessel behaviors around the maximum. This is based on two joint works with Weitao Zhu.