Laws vary a lot among countries, changing from very liberal in the US to dramatically painfull in the UK. In this section, I am not going to give a complete description of the laws of every country but just give a short summary of the relevant facts (i.e. can you purchase gun? what type? is carry allowed?).
Holger Reusch (holger@vmars.tuwien.ac.at) maintains a collection of law excerpts from various countries. The files can be obtained by anonymous ftp from ftp.vmars.tuwien.ac.at (128.130.39.16), directory /pub/misc/guns/gunlaw. If you need more information, you should check there.
Please realize that such informations may be not totally accurate and that you always should doublecheck with the local authorities. Neither I, nor the people who contributed to the list maintained by Holger take any responsibility for the content of such texts or any actions you may take based on said content. Check with a lawyer or your local police to in whenever you are unsure about firearms laws in the jurisdiction in which you live.
Two things seems to be the same in every country: if you want to buy a gun that requires a permit, you'd better have no criminal record, and you need to be over 18 (21 in the US for handguns).
Australia
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Laws vary highly from state to state. Some states will ask for a good
reason to buy a gun, others won't. License is required. Full-auto
weapons are illegal. Carry is not allowed.
Austria
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Need permit for handguns and number of handguns is limited to 2 (5 for
hunters). Need special permit for carry. No permit for rifles or
shotguns.
Belgium (Not verified yet!)
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Permit is needed. carry is illegal. I *think* there is no restriction
on caliber or number of guns.
Canada (This data from Greg Booth and Keith de Solla)
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Need a permit and 2 references to buy a gun. no other restriction I think.
Permit requires background check, photos.
Handguns and some semi-auto rifles must be registered and require special
permits.
Registered firearms require a good for one year permit for
transport to/from gun clubs.
Single use permits required to transport registered firearms elsewhere.
Full-auto prohibited. Many semi-auto firearms and some bolt action rifles
prohibited.
Concealed Carry Permits theoretically legal, but practically unobtainable.
[Keith P. de Solla adds:]
Bill C-68 has become law here in Canada, so there's additional info that could go into the Canadian section of the International gun laws part of the FAQ.
Finland
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Permit requires a good reason. Pocket pistols (i.e. barrel < 4") are
very difficult to get (usually for security officer). Full auto are
very hard to get (need to be considered as a collector).
France
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Permit is difficult to obtain. you cannot own more than on "war
caliber" (i.e. 9mm, .380 ACP, .40 S&W, .45ACP). Carry is illegal
Germany (Hunter information thanks to Lars Stein)
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If you are considered as target shooter, you can only have 2-3 small
caliber gun (<.32) and one large. Storage can be checked by police.
Hunters may buy an unlimited number of guns > 60 cm long.
Handguns are limited to a total of two revolvers or pistols.
Fully-automatic weapons are illegal. Rifles and shotguns
are limited to 2+1 rounds.
Hunters may only carry their gun to, from, and during hunting.
Carrying in public is illegal without a hard-to-get special permit.
New-Zealand
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Permit with picture is required and only valid for 10 years. Assault
rifle requires particular license.
South Africa
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Permit with fingerprint required. Full auto and semi auto rifles are
very difficult to obtain. Concealed carry is allowed. no restriction
on handguns.
Sweden
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Need permit and a good reason to buy firearm. Full auto
forbidden. carry is illegal. no limit on the number of guns (depends
on your needs).
Switzerland
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Permit is required but easy to get. no limitation of caliber or number
of guns. Full auto available for collectors only. Concealed carry is
legal except in 2 states.
The Netherlands
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You can ask for a permit after a year of membership in a club. Need
certificate from the club. No more than 5 guns. Full auto is illegal.
United Kingdom
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Need good reason to buy a gun (this is discretionary). Carry is
illegal. Police ensure for secure storage and use of the gun
(discretionary as well). You have to be member of a club.
USA
---
No real permit, just fill in a form at the gun store. Full Auto is
available but strictly regulated. Concealed carry is legal. Some
restriction may be brought by state/county or city laws.