We sketch the theory of London superconductors with a complex order parameter undergoing a phase transition dominated by thermal fluctuations. This theory is essentially that of superfluid helium, appropriately modified. We derive relations between the transition temperature Tc, the London penetration depth, the phase correlation length, the fall of Tc with reduced thickness in thin slabs and the associated appearance of Kosterlitz-Thouless behavior. At Hc2 thermal fluctuations drive the transition first order. Accordingly, magnetic properties will exhibit hysteresis phenomena and glassy behavior. Our analysis of recent specific heat and μSR measurements and of experiments on ultrathin slabs as well as the experimental evidence for hysteresis phenomena and glassy behavior in magnetic properties reveal that this theory describes a large class of superconductors that encompasses the cuprates, bismuthates and fullerenes. © 1992 Springer-Verlag.