Article 1 of the Geneva Convention of 1958 on the "high seas" states: "The term 'high seas' means all parts of the sea that are not included in the territorial sea or in the internal waters of a state." Article 2 states: "The high seas being open to all nations, no state may validly purport to subject any part of them to its sovereignty." The sovereignty of a state extends to the airspace above the territorial sea as well as to the sea floor and the subsoil beneath it.

Several things will come into play here.

First, in many cases, you will first need the permission of the cruise ship company itself to even have an Amateur Radio transmitter in your possession while on board (whether in use or not).  So your first step is to make sure you have written authorization to have your radio with you.

Next, besides the company itself you will need to have permission of the ship's captain in order to use the radio.  Do not assume you can simply throw a vertical outside of your stateroom and operate! You need to check with the ship's crew to determine what the specific requirements are for operating on that vessel.There can be time restrictions, placement issues - because their first priority is proper operation of the vessel.

If the ship is of non-US registry, you must obtain reciprocal operating authorization from the country of registry of the ship  (under what country's "flag" does the ship sail) in addition to being in compliance with Section 97.11. For example, if operating on a Dutch flagged-vessel, you must have operating authorization from the Dutch authorities.  When in international waters in this case, you are limited by the operating privileges authorized by that country. The ship is considered soverign territory of the country of flag, so their rules apply.

Once you have authorization top operate ship board, you still have to worry about reciprocal operating privileges where the ship physically is at any point in time, including territorial waters,

When an FCC licensed amateur is operating an amateur transmitter aboard a US-registered vessel in international waters, they must follow Part 97 of the FCC rules, particularly Section 97.11.  US and Canadian licensees need no special permit or authorization other than their own FCC or DOC license as long as Section 97.11 is followed and they stay within the US and International waters.

When amateurs enter the territorial waters of a country, another tier of operating permissions come into play. If on a foreign flagged vessel in a third country's territorial waters, you fall under the communications jurisdiction of that third country - which means you need operating authorization from that country as well..

There are several "categories" that could apply.  First would be authorization obtained through a multilateral arrangement, such as CEPT or IARP.  The CEPT agreement applies to most European countries and certain overseas territories; IARP applies to certain countries in the Americas; The second means of reciprocal authoziation would be through a US bilateral arrangement wth that specific country.  Finally, your operating authorization may come directly from a specific permit obtained from the foreign country's telecom authority that regulates Amateur Radio.  Such permit geneally require an application process and payment of a fee to the regulator. 

It is worth spending some time on whether the the efforts required to operate ship-board are worth the gain.  And remember, while it is always exciting to "Be the DX" station, shipboard contacts do not count for most awards, including DXCC.

You must abide by the rules of the bilateral or multilateral reciprocal operating authority or the reciprocal permit as mentioned above.

While many countries have similar rules to those of the US, the frequencies, power limitations, identification requirements and other regulations can vary widely. We in the US often assume that Amateur Radio operators in other countries have been allocated the same frequencies that we have. This is simply not the case. Each of the three International Telecommunication Union (ITU) "Regions" have their own allocation for the various services, and they do not necessarily conform to our frequencies in the US, which is in Region 2. Amateurs operating maritime mobile must not exceed the frequency limitations for each ITU Region as authorized in the international Radio Regulations.

When operating from the US, its territories or international waters, you must meet all three requirements in Section 97.11 of FCC rules: "(a) The installation and operation of the amateur station on a ship or aircraft must be approved by the master of the ship or pilot in command of the aircraft; (b) The station must be separate from and independent of all other radio apparatus installed on the ship or aircraft, except a common antenna may be shared with a voluntary ship radio installation. The station's transmissions must not cause interference to any other apparatus installed on the ship or aircraft; (c) The station must not constitute a hazard to the safety of life or property. For a station aboard an aircraft, the apparatus shall not be operated while the aircraft is operating under Instrument Flight Rules, as defined by the FAA, unless the station has been found to comply with all applicable FAA rules." When you're in International waters, you operate under the auspices of your FCC license, but you must be mindful of the frequencies assigned to other ITU Regions. The world is divided into three "pieces of the pie." You are bound by the privileges assigned to the ITU Region from which you're operating and by the privileges as outlined by your FCC license. It's a good idea to follow your call sign with the words "maritime mobile" or "aeronautical mobile" followed by the proper ITU Region number.

If your station is operated in Europe, Africa or the adjoining waters, you're in Region 1. North and South America and the adjoining waters make up Region 2. The "rest of the world" is comprised of the countries of Southern Asia (excluding the countries of the Arabian Peninsula) as well as the islands of the Pacific and Indian Oceans.

In cases where FCC rules apply, you may pass third party traffic and make phone patches to countries with which the US holds third party traffic agreements. You may not pass traffic with countries which have not signed a third party traffic agreement with the US and there are never any exceptions to these rules. Even thoughtless violation can cost amateurs the good reputation we have earned over the years.

The territorial limits extend to whatever that particular country says they do. The US claims 3 nautical miles. Most Caribbean island nations claim 3-12 miles although there are exceptions. The territorial limits of certain Caribbean nations and other island nations whose area consists of many small islands often overlap creating a larger territorial area than one would assume.