The supplements contain omega-3 fatty acids, the polyunsaturated oils prominent in fatty cold water fish like salmon, sardines and mackerel. In many observational studies, people who regularly consumed fish two or more times a week were less likely to suffer heart attacks, strokes and cardiovascular deaths than those who ate fish infrequently or not at all.

Because patients with depression experience rapid shrinking of their hippocampus, many strategies for relieving depression focus on increasing new brain cell growth in that specific area of the brain.23 There’s now evidence that increasing omega-3 intake, especially DHA, may be an effective way of treating or preventing depression, partly by protecting the hippocampus from further shrinkage.23

They also found that taking more long-chain omega 3 fats (including EPA and DHA), primarily through supplements probably makes little or no difference to risk of cardiovascular events, coronary heart deaths, coronary heart disease events, stroke or heart irregularities. Long-chain omega 3 fats probably did reduce some blood fats, triglycerides and HDL cholesterol. Reducing triglycerides is likely to be protective of heart diseases, but reducing HDL has the opposite effect. The researchers collected information on harms from the studies, but information on bleeding and blood clots was very limited. 
A lot of the benefit of fish oil seems to come from the omega-3 fatty acids that it contains. Interestingly, the body does not produce its own omega-3 fatty acids. Nor can the body make omega-3 fatty acids from omega-6 fatty acids, which are common in the Western diet. A lot of research has been done on EPA and DHA, two types of omega-3 acids that are often included in fish oil supplements.