If someone finds a lost object he is required to return it to it's owner. This is called the mitzva of Hashovas Aveidah - the commandment to return a lost object. HaShem tells us this mitzva twice in the Torah, once in Shemos (Exodus) 23:4, "If you come across your enemy's ox or donkey going astray, bring it back to him." (The Torah talks about the your enemy's ox to stress that it is a mitzva to return a lost object even if the owner is your enemy.) and once in Devarim (Deuteronomy) 22:1-3, "If you see your brother's ox or sheep going astray, you must not ignore them, you must return them to your brother. If your brother is not near you, or if you do not know who [the owner] is, you must bring it home and keep it until your brother identifies it, whereupon you must return it. You must do the same to a donkey, an article of clothing, or anything else your brother loses and you find, you must not ignore it." This mitzva is not only for lost objects. Any time you see that a fellow Jew stands to suffer a loss you are required to take action to prevent this.
There are many laws about what to do when you find a lost object. These laws are complicated and require serious study to fully understand. Whenever possible one should speak to someone who is knowledgeable in these laws before putting them into practical application.
The following is a short summary of the very basic rules, these are not full statements of the law! They are outlines of the basic principles:
There are lost objects which you are not required to return, but, instead you may keep them for yourself. Such objects are those about which we can assume the owner has already given up hope of ever regaining. In order for us to make this assumption there are several requirements.
Beyond the Letter of the Law: Even though, as we mentioned, there are times when you don't have to return a lost object this does not necessarily mean you shouldn't return it. Frequently it is proper, and sometimes even required, to return a lost object which, strictly speaking, you are allowed to keep. When in doubt ask a rabbi who is knowledgeable in these laws.
Government Law: When there are laws from the local government which require one to return a lost object you are required to obey these laws.
© Eliezer C. Abrahamson
Introduction to Jewish Civil Law