Eating fish these days is a complicated proposition. One the one hand, certain fish have the potential to be quite nutritionally valuable.
High-quality fish are a great source of clean, easily-digestible, amino-complete protein. They can also be great sources of inflammation-fighting Omega-3 fatty acids and fat-soluble vitamins (like A and D) as well as countless valuable dietary minerals.
Unfortunately, most fish available on the market today are saturated with toxins to the point of being legitimately hazardous to our health. A huge percentage of common wild-caught fish show dangerously high levels of multiple environmental toxins (like mercury and PCB's).
You'd hope farmed fish would be safer than wild fish, but disappointingly the opposite is true. Fish farming is a dirty, problematic industry and the fish produced by aquaculture operations are often even more contaminated than wild fish.
Problems With Wild-Caught Fish
The problems with wild-caught fish are fundamentally problems with our oceans. Decades of relatively-unchecked environmental pollution have left our oceans dangerously polluted by a number of toxic industrial byproducts. While regulations in developed nations have tightened considerably, it's quite unclear what the level of environmental pollution is in upcoming industrial powers like China and India.
There's a long list of industrial chemicals we find commonly in wild-caught fish, but a few of these are particularly problematic:
Polychlorinated Biphenyls (PCB's) haven't been used in the developed world since the 1970's, but they continue to persist in our environment. PCB's have been shown to be teratogenic (interfere with fetus development), suppress immune function and disrupt thyroid balance. 
Heavy metals (mercury is the most common) are released into our environment through a number of industrial processes, coal-burning power plants being the #1 source. Even in low concentrations (levels equivalent to eating contaminated fish), mercury has been shown to have neurotoxic effects. Children exposed to EPA-determined "safe" levels of mercury show measurably impaired motor development. 
Choosing Safe Wild-Caught Fish
There are a few characteristics to consider when choosing a wild-caught fish species that will minimize your exposure to these troublesome toxins.
First is the trophic level of the species. This effectively means "how high on the food chain" a given species of fish is. As we all know, small fish are eaten by medium-sized fish, which are eaten by larger fish which are eaten by even larger fish, and so on.
Through a process referred to as bioaccumulation, any toxins in the ocean water accumulate the higher a species is on the food chain. This is why small fish (like anchovies) will always have lower levels of toxins (by weight) than large fish (like tuna) harvested from the same waters.
The second characteristic to be aware of is the average age of a species at time of harvest. The longer a fish is alive, the more time it has to accumulate toxins. Migratory fish like salmon have relatively short lifespans, while open-ocean fish like marlin can be decades-old when they are caught.
The third characteristic that determines toxin levels in a given fish is the ocean from which it is caught. Fish caught in the Atlantic Ocean generally show higher levels of PCB's and mercury than Pacific fish (although this is not always the case). Arctic waters are among the cleanest on the planet, which is why Alaskan and other Arctic fish species are often quite low in environmental toxins. 
This is a fairly complicated list of characteristics - and chances are your server or the dude behind the grocery store fish counter are not going to know detailed information about where a specific fish came from. If you're going to eat fish, the only safe strategy is to educate yourself. Stick with fish species you know to be safe (see the list below), or at least to some quick iPhone google-searching before you place an order.
Why Farmed Fish Is NEVER Safe To Eat
Intuitively, it seems as though aquaculture (farming fish) might circumvent the issue of toxin accumulation that plagues so many wild-caught fish species. Unfortunately, aquaculture is a dirty industry and toxin accumulation is often exacerbated while other new problems unique to farmed fish are added to the equation.
Farmed fish are generally fed cheap "feed pellets" of which discarded fish byproducts are a primary ingredient. The fish byproducts used in feed pellets often come from larger fish than the farmed fish would eat in the wild, so the farmed fish are effectively eating above their "natural" trophic level. The result is an increased accumulation of toxins compared to wild fish of the same species.
Studies have repeatedly shown that farmed salmon has significantly higher levels of PCB's and PBDE's (another environmental toxin) than wild-caught salmon. 
The bigger problem with farm-raised fish is that these fundamentally are not healthy animals. They're fed low quality feed and kept in relatively close quarters, making spread of disease among farmed fish quite common.
Highlighting this issue is that more antibiotics are used in fish farming than in any other type of livestock operation. Whereas antibiotic use in livestock and poultry production is regulated (albeit poorly), it is wholly unregulated in aquaculture operations.  Antibiotics are used widely and carelessly, creating a number of issues including new strains of antibiotic-resistant pathogenic bacteria (salmonella is the most notable).
As with livestock production, the antibiotics used in fish-farming accumulate in the fish themselves and remain there when the fish arrives on your dinner table. Put simply, if you eat farm-raised fish, you will also be eating some level of industrial antibiotics.
Antibiotics in our food is a HUGE issue that I'm sure will garner a lot more attention over the next 3-5 years. Why? We're learned so much over the past decade about just how integral our gut microbiome is to our health, and this field of study continues to grow at a remarkable rate. Our gut microbiome plays an indispensable role in a huge variety of processes in the body, from digestion to immune function to hormone regulation.
Any sustained disruption of the bacteria living in our guts is likely to have very real negative impacts on our health. The low-level intake of antibiotics from livestock or farmed fish certainly qualifies as such a disruption.
While it is hypothetically possible to run an aquaculture operation that addresses these issues and produces legitimately clean fish, the reality is that we can't give any farmed fish the benefit of the doubt, regardless of how convincing their marketing may be - it's simply too dirty and unregulated of an industry.
It is for this reason, that I recommend to NEVER eat farmed fish - the risks are simply too great, particularly when there are plenty of safe wild-caught options out there. (see below)
It's useful to know what unless a fish is specifically labeled as "wild" or "wild-caught", you can be 100% sure that fish was farm-raised.
Safe Fish To Eat: THE LIST
I've done a good deal of research, and if a fish is not on this list, I can't recommend eating it.
It's possible there are some exotics that I've missed, so if you want to try something not on this list, you should do some research before ordering. A good way to determine if a species is a risk is to do a google search for "<species name> mercury". If anything comes up in the search indicating mercury is an issue with that species, you should take a pass. If a fish has even moderate levels of mercury, it likely will also have PCB's and PBDE's.
Of course, the species listed below apply only to WILD fish. Several of these species are commonly farm raised (like salmon and tilapia), so be sure to make sure your fish is specifically advertised to be wild-caught before ordering.
Location of catch matters too. In general, Arctic/Alaskan will be cleaner than Pacific which is cleaner than N. American Atlantic which is cleaner than European Atlantic. Freshwater species are generally clean, but it's a good idea to do a quick google search on the specific river/lake to see if there are any issues with contamination.
- Mackerel (N. Atlantic, Chub)
- Perch (Ocean)
- Sockeye Salmon
- Coho Salmon
- King Salmon
- Shad (American)
- Sole (Pacific)
- Trout (Freshwater)
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