Republic of Artsakh (previously known as Nagorno-Karabakh) is one of those republics not recognised by many. 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 USSR dissolved, the conflict reemerged with fierce fighting going on in 1991-1994 which left many towns shattered or totally demolished. Today it is possible to visit Republic of Artsakh, but only from Armenia - you will not be able to enter Azerbaijan later on if your passport shows proof of any trip to Artsakh. There are a handful of interesting historical sites, but the main reason for any Artsakh trip should be to witness first handed this troubled slice of land.
Our guide to Artsakh.
Disclaimer: We are not political here on Globe Spots, so the reason why Artsakh is mentioned under Armenia is because you can only enter through Armenia.