International law according to the United Nations recognizes a 12 mile limit around the coastline of each country with an ocean border as their "territorial waters". Past the 12 mile limit is international waters, which any boat from any nation may sail through.
There are some countries which claim a larger boundary, as they don't feel safe with only a 12 mile buffer, or they wish to protect resources within a said limit, but there is no basis in international law to back them up.
This "Freedom of the Seas" allows each nation to be an equal on the high seas during peacetime, and for fishing and trade to take place without any maritime advantage given to one country or the other.
So the short answer is the seas belong to no one, and everyone.
No country can claim exclusive access to the oceans. According to a treaty agreed upon by all the nations called the United Nations Convention on the Law of the Sea, a country is allowed to claim waters extending to twelve nautical miles or 22.224 kilometers from its borders as it own. In this region it can restrict the access by ships belonging to other nations. In most cases though, civilian vessels and even military ships of other nations which are not entering for a military purpose are allowed free access to the seas lying within the territorial belt of another country. The seas and oceans beyond the belt of 12 nautical miles are international waters which are free for anyone to use and do not belong to any particular nation.