Surgical training in medical school residency programs has followed the apprenticeship model. The learning and assessment process is inherently subjective and time-consuming. Thus, there is a need for objective methods to assess surgical skills. Here, we use the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines to systematically survey the literature on the use of Deep Neural Networks for automated and objective surgical skill assessment, with a focus on kinematic data as putative markers of surgical competency. There is considerable recent interest in deep neural networks (DNN) due to the availability of powerful algorithms, multiple datasets, some of which are publicly available, as well as efficient computational hardware to train and host them. We have reviewed 530 papers, of which we selected 25 for this systematic review. Based on this review, we concluded that DNNs are powerful tools for automated, objective surgical skill assessment using both kinematic and video data. The field would benefit from large, publicly available, annotated datasets that are representative of the surgical trainee and expert demographics and multimodal data beyond kinematics and videos.