Hydrogen bond (H-bond) effects are well known: it makes sea water liquid, joins cellulose microfibrils in sequoia trees, shapes DNA into chromosomes, and polypeptide chains into wool, hair, muscles, or enzymes. However, its very nature is much less known and we may still wonder why O-H···O energies range from less than 1 to more than 30 kcal/mol without evident reason. This H-bond puzzle is tackled here by a new approach aimed to obtain full rationalization and comprehensive interpretation of the H-bond in terms of classical chemical-bond theories starting from the very root of the problem, an extended compilation of H-bond energies and geometries derived from modern thermodynamic and structural databases. From this analysis new concepts emerge: new classes of systematically strong H-bonds (CAHBs and RAHBs: charge- and resonance-assisted H-bonds); full H-bond classification in six classes (the chemical leitmotifs); assessment of the covalent nature of all strong H-bonds. This finally leads to three distinct though inter-consistent theoretical models able to rationalize the H-bond and to predict its strength which are based on the classical VB theory (electrostatic-covalent H-bond model, ECHBM), the matching of donor-acceptor acid-base parameters (PA/pKa equalization principle), and the shape of the H-bond proton-transfer pathway (transition-state H-bond theory, TSHBT). A number of important chemical and biochemical systems where strong H-bonds play an important functional role are surveyed, such as enzymatic catalysis, ion-transport through cell membranes, crystal packing, prototropic tautomerism, and molecular mechanisms of functional materials. Particular attention is paid to the drug-receptor binding process and to the interpretation of the enthalpy-entropy compensation phenomenon.