Smartphone addiction is a recent concern that has resulted from the dramatic increase in worldwide smartphone use. The purpose of this cross-sectional study was to evaluate psychosocial factors affecting smartphone addiction in university students. The study was performed among students at the Ondokuz Mayis University Samsun School of Health (Samsun, Turkey) on October–December 2015. Four hundred ninety-four students possessing smartphones and agreeing to participate were included. A sociodemographic data form produced by the authors and consisting of 10 questions was administered together with a questionnaire involving the Smartphone Addiction Scale-Short Version (SAS-SV), the Flourishing Scale, the General Health Questionnaire, and the Multidimensional Scale of Perceived Social Support. The questionnaires were applied in a class environment at face-to-face interviews. SAS-SV scores of 6.47% of students were “significantly higher” than the participating group mean SAS-SV score. Multiple regression analysis revealed that depression, anxiety and insomnia, and familial social support statistically, significantly predicted smartphone addiction. Further studies of smartphone addiction in different age groups and with different educational levels are now needed.