Big Eye Tuna (Ahi) is in trouble in Hawaii, I know. They announced restrictions on fishing for it just this week.
There are apparently 8 different species of Tuna or fish considered to be tunas by the fishing industry. Here is a list with the current status of each species:
Skipjack Tuna (stripe-bellied Bonito) (About 50–55% of all tuna caught)
Latin: Katsuwonus pelamis
The scientific assessment of the skipjack stock is that all catching areas are moderately exploited.
Yellowfin Tuna (About 35% of all tuna caught)
Latin: Thunnus albacares
The general concerns on yellowfin is that due to increased catches of baby-yellowfin (especially in the Atlantic, Indian Ocean and Western Pacific), the stocks might suffer on the long term.
Albacore Tuna (long finned tuna) (About 7 % of all tuna caught)
Latin: Thunnus alalunga
The stock of Albacore in the Northern Atlantic is considered to be overfished, stocks in the Southern Atlantic of West-Africa are nearing a situation of over-fishing, stocks in the Northern Pacific are fully exploited, and in the Southern Pacific they have reached their critical level.
Bigeye Tuna (Ahi) (About 8 % of all tuna caught)
Latin: Thunnus obesus
Scientists agree that the Bigeye resources are fully exploited and in some cases over-fishing are occurring. Hawaii has instituted restrictions.
Atlantic Bonito (Not strictly a Tuna) (Less than 1% of tuna caught)
Latin: Sarda sarda or sarda spp
Status not able to be judged at this point, although catch numbers and size seem to be decreasing.
Northern Bluefin Tuna (North-west Pacific + North- East Atlantic + Mediterranean) (About 1.25 % of tuna caught)
Latin: Thunnus thynnus
The Northern Bluefin is a tuna species under threat in part because it is the slowest-growing Tuna.
Southern Bluefin Tuna (Southern Pacific, Atlantic Ocean, Indian Ocean) (About 0.4 % of all tuna caught)
Latin: Thunnus thynnus
The Southern Bluefin tuna is the most over-exploited tuna species.
Tongol (longtail tuna) (About 4 % of all tuna caught)
Latin: Thunnus tonggol
There is limited data available on the volume of the catch, and the status of the current stocks, which vary greatly by season.
You can read much more detail at this site