Electric power is an essential commodity of the developed world, and is critical to the continuing progress of our technology-based society, as well as to the growth of less privileged societies. In contrast to its overwhelming benefits, there is a suspicion that the magnetic component of the electromagnetic fields (EMFs) associated with power distribution and electrical appliances has adverse health effects, especially a small increased incidence of childhood leukaemia. The possibility that environmental EMFs represent a health hazard has serious economic implications for government, the electricity industry and society, as well as raising several profound scientific challenges, including, in particular, biophysical mechanisms, experimental replication and scientific uncertainty. These challenges are explored in relation to the experiences of the EMF Biological Research Trust, a UK medical research charity which funds basic research on the biological effects of extremely low-frequency electromagnetic fields (ELF-EMFs). As judged from these experiences, at the present time there is no compelling experimental evidence that environmental ELF-EMFs induce biological responses.