How does one exactly "capture" a city in modern times?

As we know, capturing a city in the medieval era was somewhat different from capturing a city within the last 100 years. I am most familiar with the Second World War, so I will use examples from such.

So my first example will be the Battle of Stalingrad. This I believe is a great example of a siege on a city because the German army was given orders to attack a city that was being actively defended by the Soviets, no warm welcome here.

It was around September 12th that the Soviets were retreating into Stalingrad, and this is where the city fighting battles started. Obviously Stalingrad had no walls, so there was a need for a method to defend the city. But there's a catch, while Stalingrad may not have had walls on the barrier of the city, the heart of Stalingrad was basically a castle in itself. The city streets, surrounded by tall building blocks, gave troops various degrees of protection from the enemy, as well as making life difficult for armored vehicles. Bringing tanks and other armored vehicles into the city was risky because there could be any anti-armor weapon sitting around the corner or on the second floor of buildings (think bazookas for a lack of a better word). Not to mention anti tank mines, yet the Soviets took it a step further. They brought their tanks into the city using them as immobile checkpoints, having the tanks only shoot but defend vital positions. Above all this, the city was very badly damaged from shelling and bombing, so debris on the city streets complicated the situation even further for vehicles. Tanks were still used, but obviously not as effective as they could be.

So whats the main course of fighting? Infantry.

Throughout the city, each side had their various checkpoints, or strategic positions that were key to victory. It could be at the rail yard, the city grain elevator, or any other factory or building that provided a strategic advantage over the enemy.

As most city battles go, the fight for Stalingrad was very intense. In order to secure a building, soldiers had to go from floor to floor clearing out any resistance. Once the building was clear, the soldiers moved on to the next building. It was never as easy as that though, because floors of buildings were not empty platforms. It was common for soldiers to be bitterly fighting room by room. Multiply that by how many rooms were in an apartment, how many apartments per floor, how many floors per building and how many buildings in the city, and you can see why city fighting was known to take a massive amount of time and effort. If they building that you just cleared gets recaptured, you have to go through the entire hellish process of clearing a building all over again.

These buildings and other various checkpoints would be constantly changing hands, most notably the railway station which happened to changed hands 14 times in six hours. The fighting was bloody and brutal, fighting in all corners of the city (including the sewers). To keep up with the German advance, the Soviets constantly brought in fresh troops across the Volga river, so if a particular checkpoint was lost the Soviets would send troops in immediately to recapture.

One thing I personally like to reflect on is the Soviet method of "hugging" the German advance. Because the Germans worked closely with their support units such as artillery and ground attack aircraft, the Soviets fought as close to the Germans as possible to reduce effectiveness of German fire support. This backs up my city stronghold/castle idea, because German support would not want to risk killing their own men to try to kill Soviets nearby. A clever method to try and avoid German siege weaponry.

I hope this shed some light onto how city fighting goes, and how one major warring power would go to seize a city. The plan was always to capture the city quick, but that rarely happens when the attacking force meets resistance.

Now on to obtaining possession of a city.

Just like in any battle, there is a line of offense. This means that across this line, the enemy is in front of you (hopefully) and your allies are behind you. So therefore, theoretically, to take over a city you should push your enemy all the way out of the city limits. Sometimes the enemy would be forced into small pockets of resistance which were hard to clear out because the enemy knows there is no where to go so they will fight to the end (or surrender). Nonetheless, as stated above typically a city is considered captured when there are no more enemy forces left in the city. So your friendly troops should be able to walk around the city without being ambushed and killed. If not, then there must be some remaining enemy troops around that don't have enough force to recapture the city but will still cause harm to your allies, so they need to be cleared out.

So we captured a city and now we have to occupy it! What do we do?

While having a soldier every couple feet across the city isn't an option, militaries usually sent out patrols that kept order in the city. Look at towns in occupied France back then for an example, while Germans didn't typically keep whole armies in cities they usually stationed soldiers in order to keep up the military presence. But usually occupying is not that easy. Sometimes the city could be completely welcoming to the occupying forces, and such the military presence wasn't needed at an all time high. But in some places the occupying forces could meet some internal resistance. In that case there are many ways of dealing with the resistance, either by fighting them off as they come by, or hunting them down as was done by the Gestapo and SS. There is no real checklist that ensures the city is yours, a "good occupation" would mean the city and it's civilians are functioning as they did before, but now the enemy soldiers are long gone.

I hope I helped you understand your question. I'm an amateur at all this history stuff so please feel free to correct me where I'm wrong!