purchasing and registering a car in Europe

Please, Europe is not a country.
The rules are different in each country. If you asked the question about how to register a car in France, Germany, Spain, or Moldova, the answer to all these questions is different. The few countries of which I have any knowledge this subject all require you to be a (legal) resident of that country with an address in the country. Your address is on the registration documents, and the documents usually require updating when you move.
There are also other legal requirements, some sort of regular vehicle road-worthiness and safety check is normal; called "MOT" in the UK, "TÜV in Germany, "MFK" in Switzerland. Insurance is required. Plus tax to be paid.
In short, you have to be a legal resident.

What about the option of a long term 12 month visa in say France or Germany?

If you have a residence visa, then you are legally a resident, so of course you can register a car in your name at your legal place of residence in that country.
How long are you planning to spend on this trip? Countries have limits on how long you can stay as a tourist. If you want to exceed these limits you need to get a visa and then you become a legal resident. The time limit varies with country, but for the Schengen Area countries the limit (for non-Schengen/EU citizens) is "90 days in any 180" and this limit applies for all Schengen countries collectively.

Another issue is your Driving Licence.
As a tourist you are allowed to drive on your home licence (possibly requiring an IDP "International Driving Permit" as well) for a limited period. After this period you need to get a licence from your new country of residence. This may require taking a driving test. The period is typically 6 months or 1 year.

Most would go into a longer term lease for periods of over 3 weeks. There are benefits to such a plan and certain rental companies (like AutoEurope) can handle licensing and insurance smoothly..
This is not a question that comes up that often because of the maximum 90 day period out of 180 days someone can be in the EU without a Visa under the Schengen Agreement. Unless someone goes to college or teaches in Europe, visa's are just very difficult to obtain.
If you went with a new U.S. spec. BMW or Volvo, they handle all licensing and insurance while the car is in Europe--prior to shipping back. Right now, their overseas delivery programs are about the best. Due to U.S. regulations, you just cannot buy a car in Europe and ship it back to the U.S.

I suggest starting at the beginning and address other issues, a car is a relatively minor tactical matter. What is your plan for staying a year in Europe? Do you have a job there? Where? In the Military? Do you already have a Visa or other means of being in the zone for 12 months? (Dual Citizenship for example) What is you itinerary or plan? Your question about a long term visa concerns me, if you had a plan, then for a departure in 3 months, you already should have one or an alternative. You cannot just show up and stay for a year in any country.

All that makes a difference, could be you don't need a car if the plan is to be in a major city most of the time, short rentals or a lease of a few months might suffice.

We bought a car to use in Europe a long time ago when we were on sabbatical in Germany. We sold our car before we left and wanted a new car when we returned. We worked through our local Volvo dealer to negotiate a price which was about 10% less than the same car in the US. It was set up converter wise (and other things too) to conform to the US market.

We got other incentives for free with the factory purchase too such as insurance for six months, license, hotel in Gotheburg, free international ferry to and from Sweden, one and 1/2 free airline tickets on SAS, free shipping and insurance to a home port. The program was called Tourist and Diplomatic Sales.

When we returned home we had to wait one month for our car to arrive. We paid no US sales tax on it because it was considered a used car in California. We drove that Volvo for 23 years. It was a great program for someone needing a car for the military, sabbatical or temporary employment in Europe. We used our car for weekend trips every week to explore the villages and hard to get to places near us. We took it on some longer trips to other countries too as breaks allowed(6,000 miles). There were 3-4 of us using the sedan as our older son went back to the US to university in the fall.

Bonnie you ask "What do other travellers do who go to Europe long term?" The answer might be different depending on where YOU are coming from, US or Australia or somewhere else. Please provide more information.

The thing is traveler's can't go to Europe long term unless they have dual nationality or a specific visa. If those are in order, they buy a car like anyone else. Unless you are carrying thousands and thousands in cash, you'd need a bank account, which in turn means you need to be a dual or have a visa to stay over the 90 days.

Europe has a lot of public transport. are you sure that you NEED a car?

Consider writing the largest military base in the country asking them hoe people register cars in that country. state. county.whatever.

I think it's pretty standard for your drivers license to be good for a year. After that you have to get a local license. I'm sure that changes by country but the few I've checked were one year.

The max time for a lease is six months, min is 17 days. You might be able to get a short term lease when you need one but I don't think you can lease for an entire year.

I agree with the others, more information would be helpful on what you're planning to do. To begin with, what is your nationality? That will have a bearing on the procedures for buying and registering a vehicle. Which country will your residence be in? "Other travellers" typically don't buy a car, since they can only remain in the Schengen zone for a maximum of 90 days.

Thank you everyone. It seems there is information that I should have provided in my initial question. I feel my question has been answered however so thank you. I am Australian and have my own migration law practice. This means I know the visa requirements very well and am eligible for a 12 month visa in a choice of several European countries due to self employment, cash at bank and possibility of university enrolment for my husband on an exchange program also. Unlike many we do not plan a year in advance and find our schedule right now suitable to our family and current circumstances. The details of what we are planning to do and the requirements of the Schengen visa are known to me.

Thank you all for your advice. I will be more careful next time with my questioning and providing more information at the outset.

There you go: once you have the visa, and you know the requirements for that, you'll be able to buy a car because you'll have an address to show. It's the same for selling the car at the end of the year.

Thanks for the additional information as that clarifies the picture.

Another thought that occurred to me is that you may want to consider obtaining an International Driver's Permit for each driver, to use for the time you'll be driving with your home D.L. The IDP is compulsory in some countries and is valid for a year.

Car ownership, registration, tax and insurance is still an area within the EU that is not remotely uniform. But it is changing and new directives that were issued in 2014/16 are coming into force. And one of the main areas of concentration is around this issues of registering/taxing/insuring cars in one state (the cheap one, of course), while it fact it is being used primarily outside the state.

Note that breaching these laws can be very expensive - one gentleman has already had to pay a fine for €15,000 for driving a Swiss registered car (owned by his son-in-law) across the border into the EU to do some shopping!

If you are going to buy rather than rent a car make sure you fully understand the new regulations.