If we can farm catfish, a freshwater fish, can we not farm saltwater fish? How about crabs & lobsters & shrimp?
not sure about crustaceans, but marine fish are quite widely farmed in big floating cages or pens made from floating/anchored nets - the species farmed this way tend to be ‘premium’ varieties; tuna, salmon, etc.
Molluscs such as oysters and mussels are extensively farmed too; the latter being seeded ontoropes or chains that are left to hang vertically down through the water.
Huge numbers of shrimp and salmon are farmed. There are concerns over the environmental/health consequences of this (e.g. concentration of PCBs in farmed salmon), but it certainly is possible.
Farming of marine fish is more recent than that of freshwater, but it is being done. There was a speaker at the 2005 National Marine Educator’s Association meeting about it; one of his business ventures is here: http://www.bizjournals.com/pacific/stories/2000/04/24/daily11.html
Surprisingly, Cates International (his company), doesn’t seem to have an easily findable website! 
Anyway, some marine fish and shellfish are farmed, particularly salmon, shrimp, and bivalves. Some fish haven’t really been experimented with and we haven’t figured out how to farm them yet. You asked specifically about crabs and lobsters, and aquaculture (water-farming) of crabs has been tried and met with some failures. According to this page, lobster farming is occurring in Monterey, CA and Prince Edward Island in Canada, but the cost and the length of time it takes for a lobster to reach marketable size are obstacles to success.
As an aside, farming oceanic species is sometimes called mariculture, a subset of aquaculture, because aquaculture is more frequently associated with freshwater.
wevets
“Farming of marine fish is more recent than that of freshwater”
Pre contact Hawaiians are thought to have been farming marine fish as much a 1000 years ago.
http://www.hawaiiaquaculture.org/q&a.html
Sadly, much of this craft has been lost and the yields of the past are not attainable with todays technology.
Cedar Key, Florida has been farming clams for quite some time. They are quite good too, not that I’m a clam expert. It was a program to help the local fishermen who were suffering due to lack of mullet and oysters. It has been wildly successful.
It looks as though we are going to find out:
http://www.alaskareport.com/crabs.htm
This site is a good source of information regarding commercial fishing throughout the state of Alaska.
The world’s first large-scale shrimp farm was right here in Panama City, FL from 1968 until 1980. In his book Memoirs of a Shrimp Farmer, John Cheshire describes the enthusiasm and efforts of a small group of locals and Japanese businessmen whose hard work and determination couldn’t overcome the pitfalls of operating a successful shrimp-farming venture.
Unagi eels are farmed extensively in Japan, particularly around Hamamatsu in Shizuoka Prefecture.
There’s extensive farms for different kinds of shellfish, cephalopo-(uh, how do you do the plural? anyway, squid and octopus) and sea fishes on the northern coast of Spain.
Cultivated pearls are from oyster farms. If my information is correct, Majorica’s first farm off the island of Majorca was the first.
Not really. That project is breeding immature crabs for release into the wild, much the way a hatchery releases young fish into the wild to mature on their own. That is much different than aquaculture, where the fish are hatched, grown and harvesting all within the confines of the aquaculture facility.
It is interesting to note that commercial farming of fin fish (salmon, cod, halibut, etc.) is illegal in Alaska.
Chinese freshwater aquaculture is more than 2,000 years old and perhaps as old as 3,000 years (see this link and and this one).
Let’s not carp on and on about who did it first. 
This discussion would be incomplete without mentioning Chuck Hesse, owner of the world’s only [ulr=“http://www.seafoodbusiness.com/archives/99mar/seafood%20star.html”]successful conch farm. Saltwater univalve farming – it’s the future!
…okay, so this discussion is still clearly incomplete. But still: conch!