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With more and more people going into space in future it is probable that crimes will be committed sooner or later.
Who will have jurisdiction on prosecuting such crimes? And who will be able to define what is the crime (let's assume that theft and violence against other person will be universally regarded as criminal behaviour, but there are lot of cases when some actions are regarded as crime in one society and are perfectly legal in other societies.
The answer depends partly on your definition of "crime" and whether international law is suitable for dealing with criminal issues. Adopting the prevailing viewpoints, however, legal issues related to activities in space are addressed by a set of international treaties, most notably the Outer Space Treaty of 1967 (OST), which has been signed and ratified by all "space-faring" nations. Under this treaty, countries have agreed essentially to be responsible for all activities in space that are conducted by their respective "entities", including governmental and non-governmental entities and private individuals. So if a citizen of Country X causes harm to a citizen of Country Y in space, then (assuming both countries are signatories of the OST) Country X has an obligation to restore the damages to Country Y.
What this means in terms of specific "crimes" committed by specific people against other people is largely an untested question. As someone else mentioned, there are more specific agreements between countries for specific cases, such as the Intergovernmental Agreement of the ISS.
Even though the OST addresses non-governmental entities, it is largely geared towards governmental actors in space. For example, if the US launches a rocket and it accidentally falls on, e.g. the Palace of Versailles, the US would probably be shelling out a large reparation payment to the French Republic. Countries generally supervise and restrict their own citizens from launching things in space, unless they insure those activities. For example, rocket launches conducted by commercial companies are almost always insured against third-party damages.
Your question, however, speaks to a big open hole in space law. If private individuals will start to operate and even live in space, Earth-bound governments may no longer wish to remain responsible for their actions. For example, one could imagine that a country might rescind the citizenship of a "rouge" space-traveler, precisely in order to avoid liabilities caused by this "space pirate". Countries would probably still be expected to provide protection and assistance to their own space-based citizens and assets against such pirates, but prosecuting the pirates criminally (or under an international legal framework) might be near impossible.
To gain deeper insight, note that the OST treaty and much of space law is largely based on long-standing customs and treaties concerning activities in international waters (e.g. Convention on the High Seas). If you think about it, international waters are very similar to space in terms of jurisdiction. Every vessel is normally required to operate under a certain country's flag, and that country then assumes responsibilities for certain liabilities. While there are certainly some pirate vessels (even today), usually the actions of those vessels are traced back to a particular country, and international pressure is applied to that country to control the pirates' actions and perhaps in seeking reparation for damages. Many wars have erupted or escalated precisely because pirates from a particular country committed crimes against ships of another country.
The Wikipedia page for Space Law is also a good starting place for learning more about these issues.
Nobody has Jurisdiction, because there is no governing body. Historically Captains have full power of command/judicial proceedings in/on their ships.
From Wikipedia Jurisdiction (from the Latin ius, iuris meaning "law" and dicere meaning "to speak") is the practical authority granted to a formally constituted legal body or to a political leader to deal with and make pronouncements on legal matters and, by implication, to administer justice within a defined area of responsibility. The term is also used to denote the geographical area or subject-matter to which such authority applies.
In theory, the Moon Treaty turns everything over to international country, however, none of the spacefaring countries ratified the treaty. Thus, it is unlikely that this will be the case. There are two likely possibilities which we can predict now, based off of two real world examples.
Most likely, the laws will most likely follow the same laws as shipping, and international airplanes, namely, that the host country of the ship is responsible for the laws of the ship, and the people are somewhat subject to their native country. Of course, these are untested, as there hasn't been any private enterprise.
A second possibility is that the rules of the International Space Station will be followed. ESA has a great write up on this, a few quotes of interest:
The Intergovernmental Agreement allows the Space Station Partners States to extend their national jurisdiction in outer space, so the elements they provide (e.g. laboratories) are assimilated to the territories of the Partners States.
The basic rule is that 'each partner shall retain jurisdiction and control over the elements it registers and over personnel in or on the Space Station who are its nationals' (Article 5 of the Intergovernmental Agreement).
You can read the complete agreement from the US State Department, take a look at Article 5.
No one will have jurisdiction since there is no sovereign territory in space. It is only when you return to someone's sovereign territory that you enter someone's jurisdiction. This gives rise to the occupation known as bounty hunting... This is no different than how it was when there was unclaimed terrestrial territory.
It is important to note that your right to self-defense is a natural right and not granted by any sovereign nation. So take some guns with you just in case.
Because space is not the sovereign territory of any particular country the jurisdiction likely falls to the UN and by extension the International Criminal Court.
In some cases, such as murder, countries claim sovereignty over their citizens outside there territory; so a US citizen would be subject to US laws. Also space exploration requires space ships and countries would likely try to exercise their laws over ships flagged in their country much like seafaring vessels.
