There are three main types of research we can do to try to find out whether EMFs cause disease: epidemiological, theoretical, and biological.
Epidemiology is the study of patterns of disease in populations. Epidemiology aims to discover if there is any statistical link or association between exposure to EMFs and disease in actual human populations. It was through such studies that concerns about magnetic fields were first raised in 1979. See an introductory tutorial on epidemiology.
The strength of epidemiology is that it looks directly at human populations. However, all it can ever do is observe statistical associations. It can never completely eliminate all the many other factors that determine whether people develop diseases or not, and so it can never conclusively prove whether a particular disease is caused by EMFs or not. More on what epidemiology does and does not show.
Around 20 epidemiological studies have now been performed looking just at a possible link between childhood leukaemia and EMFs. Numerous other studies have looked at other diseases. Some of those studies found no association with magnetic fields, but some have found associations, and consequently research continues until a clearer picture can be achieved.
With electric fields, the position is clearer: there is very little epidemological evidence suggesting they are a cause of childhood cancer, and quite a lot suggesting they are not.
Theoretical research looks for a plausible mechanism that can demonstrate how the fields could interact with living systems. A variety of theories have been put forward over the years, but no such mechanism has been established that would operate at the levels of field found in homes or near power lines.
An important test of any proposed health risk is biological research: laboratory research actually to observe the effects of EMFs on cells and tissue. There have been many hundreds of these studies reported, and scientists examine them for robust results which can be successfully repeated in different laboratories.
In over 20 years of research there have been no such well-established reproducible results. The evidence from the laboratory suggests that low level EMFs of the type experienced by the public do not cause the diseases that have been claimed. Of particular significance is the fact that standard experiments on mice and rats of the type used to establish carcinogenicity of other agents have come out negative for EMFs.