I am getting married to a US citizen next month. Can my husband adopt my 20-year old daughter?

Your daughter is of legal age and emancipated. Thus there is no practical reason for your husband to adopt her. She can certainly maintain a father-daughter type of relationship with him if she desires. She can also abstain from having any such relationship with her biological father if that's what she wishes.

Why would your fiancé want to adopt your daughter, since she is an adult now? Is this for legal matters pertaining to heirship? Or just wanting to do something "nice"? If he wants to make certain she is provided for after he dies, he can certainly list her in his will.

Or, are you hoping that if he adopts her, she becomes a US citizen? It does not work that way. Call the INS and ask them. She would still have to become a citizen the way all immigrants become a citizen. Immigration at USCitizenship.info: INS, US Immigration, Citizenship (http://www.uscitizenship.info/?ad=adwords&keyword=ins)

I am getting married to a US citizen next month. Can my husband adopt my 20-year old daughter?
Absolutely! The laws are different from state to state, in any case adoption is allowed once you're married.