Skin is the largest organ of the human body, and it offers a diagnostic interface rich with vital biological signals from the inner organs, blood vessels, muscles, and dermis/epidermis. Soft, flexible, and stretchable electronic devices provide a novel platform to interface with soft tissues for robotic feedback and control, regenerative medicine, and continuous health monitoring. Here, we introduce the term "lab-on-skin" to describe a set of electronic devices that have physical properties, such as thickness, thermal mass, elastic modulus, and water-vapor permeability, which resemble those of the skin. These devices can conformally laminate on the epidermis to mitigate motion artifacts and mismatches in mechanical properties created by conventional, rigid electronics while simultaneously providing accurate, non-invasive, long-term, and continuous health monitoring. Recent progress in the design and fabrication of soft sensors with more advanced capabilities and enhanced reliability suggest an impending translation of these devices from the research lab to clinical environments. Regarding these advances, the first part of this manuscript reviews materials, design strategies, and powering systems used in soft electronics. Next, the paper provides an overview of applications of these devices in cardiology, dermatology, electrophysiology, and sweat diagnostics, with an emphasis on how these systems may replace conventional clinical tools. The review concludes with an outlook on current challenges and opportunities for future research directions in wearable health monitoring.