I work in the industry and therefore feel qualified to answer this question.
For cards bearing a chip; if you need to use a PIN in your home country, then you will be required to use a PIN overseas.
The preferred method of authorisation is embedded into the chip in your card.
All terminals that comply to the EMV standard will request a PIN to authenticate the transaction if the card is configured as 'Chip & PIN'.
If your card is configured for 'Chip & Signature', then a PIN will not be required.
The only exceptions are in the US (and a limited number of other countries) where a proportion of retailers have still not implemented EMV terminals at their Point of Sale.
In these cases, EMV chip & PIN cardholders are required to swipe and sign, and more often than not, are requested to show photo ID.
If your card does not have a chip, then no PIN will be required anywhere worldwide.
However, you may find that your card is simply not accepted or that the terminal or machine is unable to process it.
One note for Germany in particular, credit cards are not as widely accepted as in other European countries.
Debit cards can be used almost everywhere, but you may want to plan on carrying enough cash to cover in case you find yourself in a place where you cannot use credit cards.
As mentioned in the comments to the question - vending machines in the countries you are planning to visit, as well as public transport ticketing machines will only accept EMV chip and pin cards and frequently, only EMV debit cards are accepted.
In some cases, a ticket machine may be the only way to purchase a ticket and you may need to ask the assistance of a local to buy a ticket on your behalf.
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I have used a US debit card in Czech Republic couple of years ago.
I had to enter PIN to withdraw from ATM, but for paying in shops, a signature was required instead.
I believe this is related to the fact that the card didn't have a chip (like my Czech debit card does), only a magnetic strip.
The other kind of cards I have experience with are Czech debit cards.
With new ones, they not only have a chip, but are also contactless, which means you don't have to enter PIN (or write signature) if you're paying less than 500 CZK.
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It depends on your card capabilities.
If it is EMV compliant with a chip then it may happen very probably across all Europe.
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I wouldn't recommend only having a magstripe & signature card in Europe as, in some places, the card reader won't accept magstripe or the magstripe reader is covered.
Chip and Signature will work in a lot of places that accept the card.
Keep in mind, though, that in some countries VISA and MasterCard acceptance is poor and AmEx even worse.
In the Netherlands, for example, Maestro and V-Pay are dominant (a European standard) or, in Germany, some places only accept the national standard girocard.
Although slowly it's changing, if you shop in the big chains, you probably won't have a problem.
And, as others said, some credit card readers may require a PIN to work.
The German Railway, for example, recently changed its ticket machines to prompt for a PIN, as customers have raised security concerns.
In some countries contactless payments are getting more and more popular (Especially in eastern Europe, where in some countries it's an exception not to be able to pay contactless, but sometimes also in Germany, UK etc.).
This means you can easily get along with Apple- and AndroidPay.
You may consider setting it up for your card and use your Phone for payments whenever you see the contactless logo to avoid complications.
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Assuming you are American, your credit card (and maybe your ATM/Debit card) will either be a chip & signature card or and older-style swipe and signature card.
For ATM cash withdrawals, your ATM card + PIN will almost certainly work fine.
You may be able to use a credit card for cash advances this way as well; contact your credit card company for the PIN.
Note that fees for withdrawals with your ATM card will be substantially lower than for with your credit card.
This is your most reliable means of payment (many merchants do not accept credit cards), so I HIGHLY recommend you keep some cash handy at all times.
For chip+signature cards, you will be able to pay just fine for merchants with manned terminals, with no PIN required.
Sometimes merchants will not know they need to ask for your signature, but I've never had a problem with the charge going through anyway.
You will NOT be able to pay at automated kiosks, such as many autoroute toll booths in Austria and train/mass transit ticket machines.
You MUST have cash on hand to pay at these machines.
For swipe and signature cards, you will likely be successful at higher end or more touristic restaurants and also at most hotels, but I wouldn't count on the card being usable most other places.
Source:
American living in Europe and using both US and Euro cards extensively in Austria and Germany.
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We are headed to Austria, Germany and the Czech Republic in a couple of weeks and heard we may be required to use a PIN along with our credit cards. Is this true?  I don't even know what my PIN is!
Will we be required to use a PIN associated with our credit cards in Central Europe?
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<germany><payment-cards><austria><czech-republic>
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If you need to use a PIN in your home country, this will also apply abroad, but it will not be required if it is configured for 'Chip & Signature'. While vending machines and automated kiosks will require a pin, many shops that accept cards will allow you to sign. However, this is not always true though, so it may be worth organising payment with your smartphone and cash withdrawals with an ATM card + PIN, or contacting your credit card company for the PIN. Be aware that many countries have poor acceptance of Visa, MasterCard and especially AmEx. Using a swipe and signature or chip and signature credit card may be a problem in Europe where chip and PIN cards are the norm and debit cards tend to be preferred to credit cards. Many outlets do not have the hardware to process swipe and signature paments, especially where payment is made through vending machines. Withdrawing cash from ATMs should not be a problem, however. 
Credit card acceptance is poor in some countries in Europe, where debit cards are preferred. Chip and PIN most common, problems may be encountered with swipe + signature cards as the terminals cannot accept them. Contactless payment is becoming more popular so Google and Apple pay can be used at these terminals. Cards should be fine for cash withdrawals at ATM. Debit cards are the norm, especially in Germany. Vending machines and other outlets where payment is automated will not accept swipe and signature cards.