DVD's & CD's are generally made of poly-carbonate plastic covered by an aluminium coating.
DVD's & CD's should be placed in the general waste bin.
The cases for both can be placed in the household recycling bin, remove paper sleeves and any soft plastic lining where possible.
Instead of disposing of suitable DVD's and CD's once you are finished with them, why not give to friends or family or donate them to a charity shop for someone else to use?
This is particularly relevant for music, films or box sets.