Many things have existed that did not have an owner, sometimes called the commons. The term "commons," however, is also often used to mean something quite different: "general collective ownership" - i.e. common ownership. Also, the same term is sometimes used by statists to mean government-owned property that the general public is allowed to access. Law in all societies has tended to develop towards reducing the number of things not having clear owners. Supporters of property rights argue that this enables better protection of scarce resources, due to the tragedy of the commons, while critics argue that it leads to the exploitation of those resources for personal gain and that it hinders taking advantage of potential network effects. These arguments have differing validity for different types of "property" -- things which are not scarce are, for instance, not subject to the tragedy of the commons. Some apparent critics actually are advocating general collective ownership rather than ownerlessness.
Things today which do not have owners include: ideas (except for intellectual property), seawater (which is, however, protected by anti-pollution laws), parts of the seafloor (see the United Nations Convention on the Law of the Sea for restrictions), gasses in Earth's atmosphere, animals in the wild (though there may be restrictions on hunting etc. -- and in some legal systems, such as that of New York, they are actually treated as government property), celestial bodies and outer space, and land in Antarctica.
The nature of children under the age of majority is another contested issue here. In ancient societies children were generally considered the property of their parents. Children in most modern societies theoretically own their own bodies -- but they are considered incompetent to exercise their rights, and their parents or guardians are given most of the actual rights of control over them.