Most times, I will answer "Thank you. It's very nice to meet you as well." It depends really on how polite you are trying to be. "You, too" usually sounds insincere to me (except when I'm dealing with someone whose first language is not English - then I understand). In business in the US, someone will say "Nice to meet you," when they shake hands. (and when shaking hands in the US, always shake strong - don't give someone a limp hand). If they spoke first, then while we are shaking hands, I would answer "Thank you. It's very nice to meet you as well." When you are parting after meeting the first time, you might repeat the "Nice to meet you" in your farewell. Sometimes this also has a handshake, but not always. If we are parting, and someone says "Nice to meet you," In a business context, I would normally respond with "Good to meet you as well. I look forward to doing business with you." If it is friendly only, I might say "Nice to meet you, too. I'll see you soon." Of course, the responses may change a little bit based on whether I really will see the person again, or not, but that is basically how I will respond to such a greeting.

Clarify one more after reading some responses. Not only does context matter, but it is very important what kind of image you are trying to project of yourself. How you respond will always make an impression on the person you are speaking to. Some have pointed out that "You, too" sounds insincere or lazy. That's exactly the impression it makes on me when someone responds that way. Some people may not put much importance on a greeting, but when I think about it, the greeting is the first chance to make a positive impression, project something positive about myself, and let the person I am speaking to know that I am considering them important, and meeting them is an important thing to me. If I want to accomplish these things with a greeting, then "you, too" is definitely not appropriate because it does none of these things. "Me, too" is just incorrect as some have pointed out. It may be used as a joke in a very informal and comfortable situation just because it really means "It's nice to meet me, too." Saying "I know" would be the same sort of thing. But if you are trying to project something positive about yourself, or show respect to the person you are meeting, these kinds of jokes aren't made. Respond with something that shows sincerity, confidence in yourself, and shows the person you are addressing that they are worth the energy you are expending on a greeting. Respond in such a way that the other person does not wonder "Was something wrong?" If you want to make an unambiguously positive impression, and let that person know you are being sincere, and are glad to meet them, then use a more energetic response. All of that is to say that "You, too" leaves way too much room for interpretation, can project laziness, seems insencere to the people with whom you need to be sincere, and does nothing to project a positive image of yourself. The fact that we even need to debate it shows these things to be true, at least with some people. Why take a chance that the person you are meeting is one of those people?

No, use the full phrase. If you replied with "me too", it sounds like "yes, it's nice to meet me too". If you replied with "you, too" it makes no sense because the full sentence you've just heard is, "it's nice for me to meet you".

"Me too." -This phrase wouldn't work for reasons that Peachey mentioned. It sounds like you are saying, "It was nice to meet me too." which doesn't make sense. "You too." -This is perfectly normal to say to a new friend or informal acquaintance. If you can't remember any other way, you can use this just fine. For more of a formal response, see the dialog below: A- "Nice to meet you!" B- "It's nice to meet you too!" B- "The pleasure is all mine." (This means that you are really grateful to have met the person, but this can be hard to pull off.)

Hi, you can also say, "It's lovely to meet you too/ it is a honour to meet you"