IP (Internet Protocol) addresses are how computers identify each other on the internet, and every computer connected to the Internet has a unique one. If you want to know who owns a particular IP address, that question is both easy and hard to answer.
IP addresses are a string of four numbers, varying in length from one to three numerals and separated by periods. In most cases IP addresses are owned by Internet service providers (ISPs) that people contract with for access to the internet. ISPs buy blocks of IP addresses, and when you connect to the Internet through an ISP, it assigns you one of its IP addresses. It is easy to find out what ISP owns a particular IP address.
If you have the IP address you are trying to find the owner for, you can go to a database website such as http://ws.arin.net/whois/ to find the owner. Merely type the IP address (making sure to get all the numbers and periods correct) into the field provided, and you will get the owner of the address. This will almost certainly be an ISP and not a particular individual.
It is also possible to narrow down the location of a particular IP address by using an IP location service. Because ISPs buy blocks of IP addresses at a time--for particular areas--it is possible to get a general idea of where the user behind an IP address is operating. A site such as ipaddresslocation.org can give you a general idea of what town or city the IP address is from.
Finding the ISP and the general location of an IP address is as far as you can go legally. ISPs do not give out the personal information connected to an IP address (such as name, physical address) because that is illegal. If you have a legitimate legal complaint with the person using a particular IP address, contact local or national law enforcement agencies because they are the only ones who have the legal authority to request that kind of information from an ISP.
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Breaking into the computers of an ISP to find the person using an IP address is illegal and can result in prosecution and jail time.