This text explores several aspects of electrochemistry. It includes a chapter on electrochemical nuclear magnetic resonance (EC-NMR) - a powerful non-invasive technique which brings insights to both fundamental and practical key aspects of electrocatalysis, including the design of better anodes for PEM fuel cells. There is also a chapter devoted to advances in the use of rigorous ab initio quantum chemical calculations in electrochemistry, addressed to all electrochemists, including those with very little prior exposure to quantum chemistry, and demonstrates the usefulness of ab initio calculations, including density functional theory (DFT) methods, to understand several key aspects of fuel cell electrocatalysis at the molecular level. The text also discusses the most important macroscopic and statistical thermodynamic models developed to describe adsorption phenomena on electrodes, including the state of the art adsorption isotherms, which are both rigorous, and in good agreement with experiment, as well as electrochemical promotion - the use of electrochemistry to activate and in situ tune the catalytic activity and selectivity of metal and metal oxide catalysts interfaced with solid electrolytes.