Assuming United States As far as "Will they will allow you to do that?
"
Of course!
(How do you think renters get utilities?)
As far as the responsibility to pay, that's up to you.
If you are setting things up like that to prevent being swamped with bills which your girlfriend has no legal obligation to pay, that would work.
However, if something were to happen between you, and she were to move out, then she'd probably cancel all of the utilities, and you'd have to re-acquire the service in your name, which may lead to service interruptions or early-termination fees on your girlfriend's side (I'm thinking about getting a multi-year contract on something like Internet/TV).
In my case, my wife's name is on our electric bill, our cable/internet bill, but I'm on the water/sewage and trash.
It just happened like that because of who had free time at the moment to sign up for service.
It's slightly annoying, because any time I call our providers, they won't allow me to make changes to the account without my wife present.
money.stackexchange.com/users/14835/Noah
I am assuming credit scores and such are not an issue.
If you are a serious couple; probably not.
For example, my wife's name is on many of our bills but it has ended up that I pay everything.
We don't share a last name or bank accounts, so the utilities around me certainly don't care.
(I am in the United States.)
If you aren't a very serious couple, you might consider how bad it would be to have the power or gas go off after a break up.
money.stackexchange.com/users/91/MrChrister
"Is there any problem...?" There is no problem in putting her name on all bills - most companies are happy to take anyone's money, however...
Sometimes you need multiple proofs of address and "a recent utility bill" is one that comes up quite a bit in lists of acceptable proof - so you might want one in your name.
Having a mismatch between the name of the utility bill and the credit card/bank account you use to pay it isn't normally a problem - it's OK to have one in your name.
Since we bought our new house, my wife is pleased that one of the bills is in her name now for this reason.
money.stackexchange.com/users/17785/AndrewC
Alternatively, your girlfriend can now claim she has been supporting you through bill payments and make a claim for a share of the communal home, even if it is in your name - and you pay the mortgage.
Give her a rent book - make sure she has her own room, and keep the bills in your name.
money.stackexchange.com/users/17795/tomsmith
As others have said, there is no problem with doing this but if the girlfriend happens to be a non-US citizen, who might be applying for citizenship or a green card in the future, having the additional trail of paperwork is in your best interest.
Things like joint utilities weigh heavily as evidence of your relationship.
money.stackexchange.com/users/17793/Keiron
I did this exact thing for about two years - there were no problems.
In fact, there were some advantages to this approach.
For example, when my ex left I was able to get "new customer" discounts on cable internet because I had no accounts in my name previously.
Anyone who is asking for a "recent utility bill" should accept a mortgage or property tax bill in your name as well.
money.stackexchange.com/users/19402/usethedoorknob
I just bought a house.  My girlfriend is going to be living with me, but the house is in only my name. To split the expenses, we are planning on her paying all of the utilities.  Is there any problem with the home being in my name, but the gas, electricity, etc. being in hers?  Update: We are in a serious, long-term relationship and will probably be married in the next couple years.
Is it okay to put household utilities in girlfriend's name if I own the home?
money.stackexchange.com
<united-states><home-ownership><household>
money.stackexchange.com/questions/33790
money.stackexchange.com/users/17778/user17778
This isn't normally a problem. It might mean that the providers will only speak to her if there's a problem. It's also good to have multiple proofs of address, so have bills in both of your names is useful. It is up to you who is responsible for paying what. Most companies are just happy to take the money. One partner can pay some utilities and the other pay others Having a mismatch between the name of the utility bill and the credit card/bank account you use to pay also isn't normally a problem. But bear in mind if some utility providers will only allow the account holder to make changes. Having your name on the account is also useful when you require proof of address, as "a recent utility bill" is one of the items often asked for as acceptable proof. Joint utilities weigh heavily as evidence of your relationship. 
It is up to you who is responsible for paying what. Most companies are just happy to take the money. One partner can pay some utilities and the other pay others Having a mismatch between the name of the utility bill and the credit card/bank account you use to pay also isn't normally a problem. But bear in mind if some utility providers will only allow the account holder to make changes. Having your name on the account is also useful when you require proof of address, as "a recent utility bill" is one of the items often asked for as acceptable proof. Joint utilities weigh heavily as evidence of your relationship.  