Over the weekend, Turkish troops launched an aerial and ground assault on American-allied Kurdish militias in Syria. Here is what you need to know.

Aren’t the U.S. and Turkey friends?

Only to a point. As NATO allies, they are obliged to come to each other’s aid in the event of an attack. The United States is an important supplier of arms and military aid to Turkey, and has used Turkey’s Incirlik air base as part of its campaign against the Islamic State in neighboring Syria.

But relations have cooled as Turkey has taken an authoritarian turn under Recep Tayyip Erdogan, its de facto leader since 2003. Ties deteriorated further after a failed coup attempt in July 2016. Mr. Erdogan said the failed uprising was fomented by the cleric Fethullah Gulen, a former ally now living in exile in Pennsylvania. He has accused the United States of sheltering Mr. Gulen.

Since 2016, Turkey has improved relations with Russia and Iran, two major foes of the United States. Until now, all four countries have shared an interest in defeating the Islamic State, also known as ISIS, but now that the terrorist group is on the run, the conflicts are coming back to the fore. Russia and Iran back Syria’s president, Bashar al-Assad, who has held on to power after seven years of civil war. The United States has insisted that Mr. Assad must go, but he increasingly appears likely to survive, a prospect that worries Turkey.