  We study general models of holographic superconductivity parametrized by four
arbitrary functions of a neutral scalar field of the bulk theory. The models
can accommodate several features of real superconductors, like arbitrary
critical temperatures and critical exponents in a certain range, and perhaps
impurities, boundary or thickness effects. We find analytical expressions for
the critical exponents of the general model and show that they satisfy the
Rushbrooke identity. An important subclass of models exhibits second order
phase transitions. A study of the specific heat shows that general models can
also describe holographic superconductors undergoing first, second and third
(or higher) order phase transitions. We discuss how small deformations of the
HHH model lead to the appearance of resonance peaks in the conductivity, which
become narrower as the temperature is gradually decreased, without the need for
tuning mass of the scalar to be close to the Breitenlohner-Freedman bound.
Finally, we investigate the inclusion of a generalized "theta term" producing
Hall effect without magnetic field.
