Healthcare in Iceland

The health care system in Iceland is a NHS style system, which means everyone is covered by the celandic Health Insurance (sjukra.is), which is financed via taxes. There is (thank god!) no private system to speak of, except for dental care, which is not covered by the Icelandic Health Insurance.
Immigrants who have been legally residing in Iceland for 6 month are automatically covered. If you are from an EEA country can get coverage via the European Health Insurance Card or are insured directly if they have been insuread/resided in one of the EEA member states 6 month previously. They have to request form E104 (or form E106/E109/E121) from their previous insurance provider and submit it to the Icelandic Health Insurance.

For people outside of the EEA, check the link above from ECS for info.

Even for insured patients there are quite a lot of out of pocket payments, for example for interventions, such as physiotherapy, psychological care or medication (often even for life threatening or chronic conditions, such as heart medication or cancer). The out of pocket payments are paid up to a certain amount, after which the insurance will pay a higher percentage or all (see table below).

Medication payment plan for 12 month period
                                      
         Out of pocket     total       % out of pocket     % insurance
1     22.000 kr.      22.000 kr.           100%                  0%
2     31.750 kr.             87.000 kr.             15%                 85%
3     62.000 kr.             490.333 kr.       7,5%      92,5%
4              0 kr.          > 490.333 kr.       0%           100%

As a reference 62.000kr, the highest amount one might have to pay for medication in a 12 month period is ca 415€. A similar plan exists for other out of pocket payments (eg for physiotherapy)

The trade unions are an important part in the social security system. Almost everyone (85%) who is employed is in a union. Unions will often refund part or all of the out of pocket payments for medication, dental care or therapies. They also have sick day funds for after sick pay rights expire and disability funds.
See the these two links as an example: vr.is/english/sick-payfund/ vr.is/english/grants/
You can find out which union you belong to by checking your working contract or your payslip (union dues are taken automatically from the wages)