In this introductory lecture, we will review some of the basic geometric features of several models in the KPZ universality class. The lecture will primarily focus on models of last passage percolation where one considers maximum energy paths (termed as geodesics) between points in a random field. We will discuss various properties of geodesics and further attempt to offer a glimpse of several recent developments in the study of such models of random geometry accomplished via different combinations of probabilistic and geometric ideas as well as more integrable inputs. No background will be assumed.