Are there ionic solids that conduct electricity? We are taught in school that ionic substances don't conduct electricity, and when they do, it is either because they are in a molten state or because they are in solution. I understand these concepts.

Are there any exceptions? By exceptions, I am looking for ionic compounds that may not conduct electricity well, but they conduct electricity better in comparison to other ionic compounds. Additionally, what makes such a compound a better conductor of electricity?
Yes. 

Another example is VO2. Over 67 °C its crystal structure changes to rutile with a band gap of ~0.1 eV. Dopants of other substances can reduce the temperature at which it becomes a conductor.