Republic of Artsakh (previously known as Nagorno-Karabakh) is a republic recognised by few, and definitely not by Azerbaijan. It lies within Azerbaijan, but is disputed between Armenia and Azerbaijan. Mostly populated by ethnic Armenians, it was however established as part of Azerbaijan during Soviet Times. When the USSR dissolved, the conflict reemerged with fierce fighting going on in 1991-1994 which left Artsakh wartorn. Today, it is possible to visit Republic of Artsakh - but only from Armenia. You will be denied entry to Azerbaijan if they find any proof (souvenirs, photos, visa) that you have been to Artsakh.
Our guide to Artsakh.
Disclaimer: We are not political here on Globe Spots, we are just presenting things from a traveller's point of view.