The siege of Magdeburg (20 October-11 November 1806) came in the aftermath of the twin French victories at Jena and Auerstädt, and the surrender of the city marked the end of significant Prussian resistance in 1806.
In the aftermath of the defeats at Jena and Auerstädt the main Prussian field armies retreated towards the fortress of Magdeburg on the Elbe, with the French in close pursuit. On 17 October the Prussian reserve corps was defeated at Halle, losing half of its men. The survivors also retreated towards Magdeburg. On 21 October the French crossed the Elbe at two places to the east of Magdeburg and by the morning of 22 October a third bridgehead had been established. Further west Soult, Ney and Murat's cavalry were close to Magdeburg. They only just missed the main Prussian army. Murat's cavalry got close enough to Hohenlohe to demand that he surrender, but the Prussians withdrew on 21 October and began to move north-east towards Stettin and possible Russian aid.
General Franz Kasimir von Kleist, then aged 73, was left to defend Magdeburg. He had around 25,000 men to defend the city. For most of the siege the French didn't realise how many men were in the city, and a more determined Prussian commander might have taken advantage of his superior numbers to inflict an embarrassing defeat on Ney. Kleist was not the man to take that chance.
Napoleon decided to leave Ney's VI Corps and two regiments of dragoons under the command of General Klein, command of the 1st Division of Dragoons in Murat's Reserve Cavalry corps. This gave him around 18,000 men. Ney arrived at Magdeburg on 23 October.
For the first few days Ney could only blockade the city from the south bank of the Elbe, but he was soon able to build a brigade above the city, and moved his light cavalry onto the north bank. This completed the isolation of the city, but so far Ney lacked heavy artillery.
On 4 November von Kleist launched a sortie against Ney's blockading troops, but this effort was quickly defeated. On 5 November Ney asked for permission to begin a bombardment of Magdeburg. Napoleon approved this request, and siege mortars began to move towards Magdeburg.
They wouldn't be needed. What little enthusiasm for resisting a siege there may have been in Magdeburg was rapidly eroded as Prussian resistance ended elsewhere. Within days of the start of the siege Berlin fell to the French. On 28 October Hohenlohe and the main army surrendered at Prenzlau, and on 6 November Scharnhorst surrendered the last active Prussian field army at Lübeck. In the meantime Ney had carried informed von Kleist that the mortars were on their way, and fired a limited number of shells into the city in an attempt to undermine the morale of the citizens and defenders.
On the same day as the surrender at Lübeck Ney allowed one of the Prussian officers captured at Prenzlau to enter Magdeburg. With the main army lost Kleist opened negotiations with Ney. On 7 November an armistice came into effect. On 8 November Kleist formally surrendered. The French occupied the gates of Magdeburg on 10 November and on 11 November the Prussian garrison marched into captivity. Ney had captured 22,000 soldiers and 600 guns. The surrender of Magdeburg ended the military phase of Napoleon's triumphant campaign in Prussia, but Frederick William III of Prussia had escaped into Poland, where he took refuge with the Russians and refused to enter into negotiations with the French. Napoleon was forced to turn east and launch a costly invasion of Poland.
Kleist has been criticised for his lack of determination, but given the circumstances of early November 1806 it is hard to see what purpose further resistance might have achieved. The King of Prussia had fled the country, the army had been defeated repeatedly and the main field army had surrendered. There was no chance of any relief force reaching the city before the French could begin a formal siege that would devastate the city. The war had already been lost and further resistance on Kleist's part would have been futile.