The Mongol forces were expanding and marching westward, destroying any resistance. The Siege of Baghdad was brought by Hulagu Khan, another grandson of Genghis Khan and brother of Möngke. Baghdad might have been spared: Möngke told his brother to forego overthrowing the Abbasid Caliphate if they were willing to pay tribute.
The current caliph, al-Musta'sim, was not so inclined. The Abbasid Caliphate was not as strong as it had been previously: despite the jewel that was Baghdad; the Abbasids no longer possessed what was once a far-reaching empire. al-Musta'sim had not prepared militarily but believed Baghdad was strong enough to survive an attack. (He is still criticized for not acquiescing to Hulagu's demands and saving his city and people.)
Hulagu had just marched through what is now Iran, facing and overcoming such widespread opposition that Iran's agrarian potential was devastated for a generation. The mountain stronghold Alamut had recently fallen to him, and he had vanquished the breakaway Nizari-Ismaili sect called the Order of Assassins. His successes were credited with planning as well as numbers. This military push had been a couple years in the making, conscripting 10% of the Mongol population. It also included Christian warriors from Armenia, Frankish Crusaders from Antioch, and (perhaps most important for the current campaign) 1000 Chinese artillery specialists.
The siege began 29 January 1258. Hulagu had sent two columns of soldiers, one on each side of the Tigris, surrounding Baghdad. They brought up their siege engines (pictured above in a 14th century painting). al-Musta'sim sent a hastily gathered and poorly prepared force of 20,000 to leave the city and attack, but they were no match for the Mongol forces, who quickly breached dikes on the Tigris, flooding the area behind the Abbasid forces, trapping them.
And then things got worse. Join me tomorrow for what was probably the single bloodiest loss of life in the entire history of human warfare.