Over the years, mobile phones have become versatile devices with a multitude of capabilities due to the plethora of embedded sensors that enable them to capture rich data unobtrusively. In a world where people are more conscious regarding their health and well-being, the pervasiveness of smartphones has enabled researchers to build apps that assist people to live healthier lifestyles, and to diagnose and monitor various health conditions. Motivated by the high smartphone coverage among young adults and the unique issues they face, in this review paper, we focus on studies that have used smartphone sensing for the well-being of young adults. We analyze existing work in the domain from two perspectives, namely Data Perspective and System Perspective. For both these perspectives, we propose taxonomies motivated from human science literature, which enable to identify important study areas. Furthermore, we emphasize the importance of diversity-awareness in smartphone sensing, and provide insights and future directions for researchers in ubiquitous and mobile computing, and especially to new researchers who want to understand the basics of smartphone sensing research targeting the well-being of young adults.