Double-walled carbon nanotubes are coaxial nanostructures composed of exactly two single-walled carbon nanotubes, one nested in another. This unique structure offers advantages and opportunities for extending our knowledge and application of the carbon nanomaterials family. This review seeks to comprehensively discuss the synthesis, purification and characterization methods of this novel class of carbon nanomaterials. An emphasis is placed on the double wall physics that contributes to these structures' complex inter-wall coupling of electronic and optical properties. The debate over the inner-tube photoluminescence provides an interesting illustration of the rich photophysics and challenges associated with the myriad combinations of the inner and outerwall chiralities. Outerwall selective covalent chemistry will be discussed as a potential solution to the unattractive tradeoff between solubility and functionality that has limited some applications of single-walled carbon nanotubes. Finally, we will review the many different uses of double-walled carbon nanotubes and provide an overview of several promising research directions in this new and emerging field.