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Please note that UNLESS you have worked one day in a socially insured job, you will NOT get an Austrian E Card, which opens up the health service to you! You may have to rely on your EU card (ie if Ex Pat from the UK) which will NOT entitle you to everything free. Some charges may well be incurred.
The AMS or unemployment agency may insist you become registered, which means a trip to your local county health care office, usually in the same town, or definitely in Vienna/Wien. There, they will reluctantly issue you with a health care number (usually 4 digits followed by date of birth). Eventually a chipped E card will be issued to you direct.
Austria claims the unemployment benefit back from any EU country.
It is highly advisable that you take a private Austrian health scheme out until in full employment. Once you are registered, all doctors in your local county scheme and hospitals are available to you. If you are in a well paid job, you may wish to opt for higher contributions ensuring you get private room care in hospital.
I have had considerable experience of the Austrian basic national health system, and have found it to be far superior to the one I left in the UK, and probably second only to Germany and the US private health care. Be warned that once an Austrian doctor has got you in their clutches, they check for EVERYTHING! You are given an almost complete health check, and here in West Styria/Steiermark waiting times are not too bad, or non-existent!
When I have been in hospital, I was lucky to have small side wards, maximum 2 beds. Even the 'larger' wards have a maximum of 8 beds! So I would highly recommend the Austrian health system. Their food too is generally excellent! Care by staff is also perfect. Doctors highly knowledgable.

I don't know where your from , if. You are from the UK  you are covered under the UK National health system,  if you are on a part UK pension, I didn't know this when I arrived and paid into the Austrian  GKK ( notional health ) for 4 years ,  there was a 6 month waiting period to be served before any claim , fir that I took out private health cover with my insurance agent Uniqua  .
      You have to register   As a resident  at the GKK ,   
     I have been here 4 years and find the health system us fantastic compared to Australia and the UK , no appointments nessasary to see the Dr ,   Any specialists you can get into see quickly and  optician and dentist are both easy to see ,  basic dental work is free on NH , glasses are cheap here compared to $4-500 Australian ,  the Dr  will do house calls and  I have had to call an ambulance  a couple if times  and been into hospital for surgery  , no waiting time , saw Specialist on the Monday and by the following Sunday was in hospital and surgery  on the Monday  in hospital till the following Saturday , total cost €65 fir my meals off 6 days .
      Brilliant !!!/-)
PS. BTW , I was refunded the GKK payments. I had paid €4500  over 4 years :-)

Yes once registered , at GKK  they will issue the Austrian EU health card card , that covers you fir Austria ,  any where else in the EU I can get emergency health care with my UK health care EU card .

My experience is fairly limitied. When I first arrived I had to go to the hospital because of a strained ankle, I still had a german insurance and it was covered normally. I only had to pay for a Splint (Schiene I hope that's the right English word)

At the dentist I usually have to pay for the anaesthetic shot. The dentist also offers premium fillings and extra cleaning that is not supported by my insurance (GKK). Extra implants are also not supported by my insurance but I believe that's a norm everywhere.
In addition to the GKK my bank recommended a life insurance that can cover more expenses when it comes to broken bones or accidents that might make me not able to work anymore. Not sure this counts as private health insurance though.

I did not know that either...but I am not yet a UK pensioner, being only 59, and have got another 8 years of working life ahead of me. Not sure if I would be able to claim back the social contributions, as I do not pay income tax in the UK, but here in Austria. Another footnote on a similar subject, if you want to draw an Austrian pension, you have to work a minimum of 10 years full time, and for the full pension, I fear it is a full 40 years of working life before you can claim a FULL pension. Beamten or civil servants, as in the UK (which in Austria INCLUDES teachers!!) can retire far earlier, as they can in the UK...originally born in Solihull, south of Birmingham, but emigrated from Royal Leamington Spa.

I have been in hospital many times in Vienna but find the system is going down fast, I have had 2 good knee operations but under private consultant first. The normal insurance system is lacking in good health care and depends on the hospital and consultant. I have just had 2 spinal operations in the A.K.H, in Vienna and was not a success in fact worse can't walk and in worse pain. So have gone the same way as my knee operations private consultant and good hospital. The specialist doctors under the system I find good and helpful as the eye doctor I was at today.
To answer your question the private sector is much better for seroius problems with health,  dental is very good on both public and private when choose correctly. But find the skin doctors of little use and lack much knowledge, treat myself with better. As for the differance between Rhuma and Boriosis not many know. Had Boriosis for over 5 years asking to be checked so many said no such thing, when 1 doctor tested and came possitive result the treatment was wrong for long term, now am losing the use of right wrist and high pain in arm control with high cortisone, No one knows how to treat the pain?? many think it is still rhuma but no rhuma treatment functions not even at over 1000 euro per infusion.