Is there such a thing as a European SIM card?

Sort of. Every one sort of has roaming though, or per meg data charges. It's still got to be based somewhere.

The closest I've heard of is Go Sim which has a 'Europe Sim' for this use case.

Includes: a Europe SIM Card with $10 airtime

We've got the SIM for you! Our all-in-one replug sim will work in any unlocked mobile device (including iPhones, iPads and more). Simply pop out the size you need and follow your mobiles instructions on how to insert the SIM. There's no need to worry about which size to order!

Make Calls: from 25c/min
Receive Calls: from 25c/min
Send and Receive Texts: 25c/message
Prepaid Data: $0.25/MB in Europe Zone, $0.59/MB in the Rest of the World
$10 Airtime Included

Frankly, though, there's so much wifi around, I'd personally not bother even getting one if you're on holiday, but obviously everyone is different.

A lot has changed on the European cell phone market since this question was asked 2.5 years ago and the other answers are pretty much all outdated.

Since June 15th 2017, EU roaming regulations basically prohibits roaming surcharges when using a subscription from one EEA country in another EEA country. In most EEA countries, you are as a tourist easily able to get a pay as you go subscription and use it for the same costs in all other EEA countries.

Operators are however starting to get creative in their search for reasons to make use of any of the exceptions to the 'roam like at home' intent of the new regulation, some operators use bandwidth throttling as a mean to restrict data usage abroad and some operators are even disallowing roaming completely on their cheaper products.

With products, practice and pricing changing almost daily, it is impossible here to recommend any specific service.

This is not a total a Europe solution, but from the UK one might want to look at Three Pay As You Go "Feel at home" SIM Card. For 15 EU unlimited data for 30 days in Austria, Denmark, Finland, France, Italy, Norway, Republic of Ireland, Sweden, Switzerland, and as a plus the USA. I'm leaving for Europe again Sunday and haven't actually tried it, but its working fine in the US.

Germany that I visit frequently is missing so I have used and just paid for another 30 days of 7.5 GB of 4G data from O2 Deutschland. They have several data packages for 30 days the 7.5 GB 4G data is 35 EU. All their webpages are in Germany, but Google Chrome translates them on the fly. I ordered both cards from ebay and use a dual SIM phone.

Another SIM I have used is OneSimCard that has several different size Europe data packages the largest gives you 1 GB data for 30 days-kind of high at $125. These packages mostly cover EU countries plus Russia and few other countries: Austria, Azores, Belgium, Bulgaria, Canary Islands, Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Guadalupe, Guernsey, Hungary, Iceland, Isle of Man, Ireland, Italy, Jersey, Latvia, Lithuania, Luxembourg, Malta, Monaco, Netherlands, Norway, Poland, Portugal, Romania, Russia, Slovakia, Slovenia, Spain, St. Barth, St. Martin (French) Svalbard, Sweden, UK and Vatican City.

And yet another option is Truphone 9 cents per minute, text or MB in UK, the US, Netherlands, Poland, Spain, and Germany plus some other non-Europe. I actually use this SIM for one of my phones here that is used infrequently as there is no monthly charge – just use.

Some of the cheap US T-mobile plan gets you free & unlimited texting and data pretty much world-wide. Calls are 0.20c/minute. Data is technically only 2G but it often upgrades for 3G or even 4G, if that's the only thing available. Works great for navigation like Google Maps. If you have a friend that's already on a T-mobile plan you can probably tack on an extra line for $10/month. I think it's $20 one time for the SIM.

Since July 2017, there are no more roaming fee in European Union

Basically, you can use your mobile data / text messagse / calls as if you were in the country where you bought your sim card.

European Union roaming regulations apply to the 31 members of the European Economic Area; the 28 members of the EU and their outermost regions plus three EFTA member states Iceland, Liechtenstein and Norway.

Quoted from wikipedia, There is a fair limit usage :

To prevent misuse (i.e. cheaper tariffs available in the eastern members to be used constantly in the western members where tariffs are higher) a fair-use policy was mandated which would allow EEA citizens to use their phones while roaming without extra charges for business and leisure, but would still limit the use to prevent misuse and extra costs to mobile operators

So for example, if you buy a sim card in France, you can use it without any roaming free in Spain, Norway, Malta etc...

International sim is a good option, but if you need internet, I would suggest you to get a local prepaid sim. I found it much cheaper. When I went to Lithuania, I took a prepaid sim ezys (http://www.ezys.lt/en/) which I was suggested and for a bit more than 2 € I got 1GB of 4G internet! I don’t say that in other country you will find the same prices, but I assume that you can find something similar or at least cheaper than paying for every KB. So really worth checking the local prepaid sims to the country you go to as well if you need more than few KB. And if you don’t need much, so I guess Wi-Fi spots would be enough.

If you make your calls through internet, I can recommend you the German tariff "Netzclub Sponsored". After you have used the 100 +300 free MBs, you still have throttled mobile data. That's enough to make free phone calls through internet. In 2017 I tested it while I was in Scandinavia, and indeed I got no extra fees. If you cannot call your contacts via instant messenger software, you still can call landline numbers via cheap VoIP providers. After you registered your VoIP account, you can set the provider's server data in your smartphone's SIP settings.