Ohm's law is not at all universal, it just comes from the empirical observation that the current density J which arises in a conductor due to the prescence an external electric field E will usually itself be proportional to the applied electric field:
[tex]\vec{J} = \sigma \vec{E}[/tex]
Where [itex]\sigma[/itex] is called the 'conductivity' of that particular conductor, and this conductivity is 'universl' for the particuar material e.g. copper (at a certan temperature). Two generalizations are possible: sometimes the response of the material is not linear, and so the relationship is not simple proportionality; in a diode, it is possile to apply an electric field below a certain threshold strength and in a particular direction and have the output current be zero. Another possible generalization is that the electric field and the current desity do not have to be parallel, for example in crystal that has its own preferred directions for conduction channels.
