In most cases your employer will be responsible for providing your insurance. But insurers may have their own requirements, particularly in the case of liability insurance.
What’s more adequate insurance is an important element in managing risk. Different insurers have different requirements, and you should be clear what these are. Most university researchers are covered by their University insurance, but insurance can become an issue in the following contexts:
Where you are engaging in research where there is or seems to be significant risk to yourself as a researcher either because of the location of the research, because of the subject of your research or the target population which you are researching.
International fieldwork may need additional clearances in terms of your institutional insurance, and so you should check with your research office or research tutor if this applies to you. That is true of any international research, but particularly relevant if your research is taking place in a country which the Foreign and Commonwealth Office advise against travelling. To get clearance from the insurer, you will probably need to set out the measures that you are taking to stay safe. It is worth noting that you may well know more about the situation in the country you intend to go to than your insurance company does. Equally, there are likely to be people in the research support department of your institution who are experienced at dealing with insurance forms, and who can help you.
Where your research might pose a risk for participants, you will need to ensure that your insurance is sufficient to cover this risk. This is generally much more of a concern for clinical trials, but if you are funded by an EU framework programme then it is important that you show that you have adequate insurance in place, and similarly, you will need to show that you have adequate insurance on applications to NRES ethics committees
Institutional insurance cover may apply differently to student research, and so you should check with your supervisor or research tutor if you have any concerns about potential risks in the work you are doing.