Armed forces belonging to Azerbaijan and Armenia both say they have the upper-hand.
At the heart of the conflict is a dispute over control of the mountainous region of Nagorno-Karabakh.
It is recognised as part of Azerbaijan, but has been controlled by ethnic Armenians since a war in the 1990s.
Tens of thousands of people died during the war and a million others were forced to leave their homes.
Other countries are concerned that the latest fighting could spill out of the region and draw in neighbouring powers, including Turkey, Russia and Iran.