People are immersed in electromagnetic fields from such sources as power lines, domestic appliances, mobile phones, and even electrical storms. All living beings sense electric fields, but the physical origins of the phenomenon are still unclear."

Source : Magnetobiology : underlying physical problems, Academic Press (3 Aug. 2002), (Other pages available at here )

It's known that electromagnetic radiation (EMR) in wavelengths which can cause tissue-heating (typically in the microwave part of the EM spectrum) can be dangerous if sufficiently intense.

EM in the ultra short wavelength range - such as X-Rays and Gamma rays - are termed ionizing radiation, and are also dangerous, being able to cause tissue damage and permanent genetic mutations.

But the nature and extent of biological effects of low-level non-heating + non-ionizing EMR are largely unknown.

The subject is under intense investigation due (in part) to the proliferation of mobile electronic devices (cellphones, WiFi etc) which radiate in the GHz region of the spectrum. Often over long durations and close proximity.

Despite the commonly accepted results, some research has been conducted to show that weaker non-thermal electromagnetic fields, (including weak ELF magnetic fields, although the latter does not strictly qualify as EM radiation, and modulated RF and microwave fields have biological effects. Fundamental mechanisms of the interaction between biological material and electromagnetic fields at non-thermal levels are not fully understood.
The World Health Organization has classified radio frequency electromagnetic radiation as Group 2B - possibly carcinogenic.

Source : Wikipedia