Epidermal electronic system (EES) is a class of integrated electronic systems that are ultrathin, soft, and lightweight, such that it could be mounted to the epidermis based on van der Waals interactions alone, yet provides robust, intimate contact to the skin. Recent advances on this technology will enable many medical applications such as to monitor brain or heart activities, to monitor premature babies, to enhance the control of prosthetics, or to realize human-machine interface. In particular, the contact between EES and the skin is key to high-performance functioning of the above applications and is studied in this paper. The mechanics concepts that lead to successful designs of EES are also discussed. The results, validated by finite element analysis and experimental observations, provide simple, analytical guidelines for design and optimization of EES with various possible functionalities.