Play any WWII game (such as the one that inspired this question, Call of Duty 2) and you will be given the option to pick up the weapons of enemies you have killed. Was this a common thing in the real world, or was it frowned upon by commanders? Was this practice influenced by the relative quality of the guns produced by the fighting nations? (e.g, Germans being more inclined to take American guns or vice versa)
There are tons of photos out there of German soldiers carrying Soviet weapons.
Some of this was undoubtedly soldiers grabbing weapons off the ground and pressing them into service; but the Germans also captured a ton of weapons during the early part of the war and later gave them German designations and issued them to their own soldiers. In the case of the PPSh-41 (the gun in the photograph above) they even went so far as to rechamber a bunch of them to fire the German 9mm round rather than the Soviet 7.62x25 that they originally used. The .30 mauser pistol cartridge was also similar enough that it could be fired in unmodified Soviet submachineguns.
US soldiers brought quite a large number of guns from both Europe and the Pacific, to the point where gun auction sites have piles of them for sale even to this day. I don't know how common it was to actually have used any of them in combat rather than merely picking them up as souvenirs. I kind of doubt it was that common but maybe someone who knows better than I do can answer that question.
Red Road From Stalingrad: Recollections of a Soviet Infantryman by Mansur Abdulin has a chapter where he talks about the guys in his unit's mania for fooling around with captured German weapons. One of his guys accidentally shot himself while handling a German gun he wasn't familiar with. They were so worried that the authorities would think that he'd shot himself to get out of combat that they just kind of bandaged the wound up on the low and didn't tell anyone outside of their immediate group what had happened. I guess you could add "potential for accidental discharge" to reasons why it wouldn't have been a good idea to mess with enemy weapons.
I hope this helped, if you have any questions I can hopefully either answer them or point you in the right direction.
There are several problems with using enemy weaponry that discourages doing so on anything more than an immediate, ad hoc basis. During WWII, soldiers carried complex weapons with a wide array of parts and pretty varied ammunition types. Getting replacement parts or even ammo for a captured weapon would be difficult to do at an institutional level.
American soldiers were happy to take some German weapons as souvenirs. While not particularly academic, Band of Brothers does a good job depicting American soldiers hunting for Lugers, Hitler Youth knives, or anything else distinctly "Nazi." Soldiers and Marines in the Pacific did the same with Japanese swords.
The Germans did operate a few captured bombers, but it was really more of a curiosity than a major part of their war effort.
Similar question:
I'm an American infantryman that happens to spy a Luger in the holster of a nearby dead German officer. What's my process getting this home as a souvenir? Do I have to hide it? Is there a special form I can sign? Can I just go to my CO and say "Hey, I'm taking this"?
Finland's history is interesting in this sense because their nation shared the same rifle as their Russian neighbors; the Mosin Nagant rifle. The Finns however never manufactured any receivers of their own. They either purchased or captured equipment, and then re-issued or arsenal refurbished them. While a majority of their rifles were purchased, it is very common to come across a Russian issued weapon that was captured and refurbished by the Finns. Many of these are still for sale today in the US as surplus. A couple good sources for Finnish Mosin Nagant rifles as well as other nations use of this rifle are http://www.mosinnagant.net/finland/finnfaq.asp and http://7.62x54r.net/MosinID/MosinIDII.htm#Finnish
I happen to have a rifle that the Finns Overhauled; a US manufactured Westinghouse M91 Mosin Nagant (there's some info on the history of this rifle in the album descriptions).
In addition to German capture and re-issue of weapons, the Russians did the same thing and still to this day, Russian WWII captured weapons are still for sale. Russian captured 98k rifles are in particular very common, though they were not typically re-issued immediately. They were typically were taken to an arsenal, had the metal reblued, the stock covered in a Russian shelac finish, sight hood and cleaning rod removed to be scraped for steel, Nazi markings were often peened out, and parts were force serial number matched using electro pencils. The same practice was also common on other German weapons. Typically these weapons were used post war and issued to other Soviet nations.
This website details some interesting information about the M1 Carbine including discussion of German use of the US issued M1 Carbine.