I Am Looking Forward To (Meet / Meeting)? You?

"I am looking forward to meeting you" Is the correct answer.

The verb is 'to look forward to' = 'to anticipate' (transitive = requires a direct object).

So, the direct object should be the gerund (noun) form of the verb 'to meet', i.e. meeting.

In contrast: "I'm waiting to meet you".

Only #2 is correct. The phrasal verb 'look forward to' must be followed by a noun or, as in this case, a gerund, which is a noun equivalent.

Best wishes, Clive

The actual problem in this case is, that "to" is a preposition here and NOT part of the infinitive! Therefore the Gerund is required.

After verb+preposition, only the Gerund can be used:

I'm fond of doing sports. I'm looking forward to seeing you. 

After some verbs, there's only the infinitive with to that can be used:

I wanted to see you. 
-> Here, "to" is part of the infinitive and no preposition. 
("I wanted seeing you" is not possible.) 

And after a few verbs, either a preposition or an infinitive is possible;
In some cases, the meaning doesn't change, while in others it does change:

a) I started to walk or I started walking. (same meaning) 
b) I stopped to smoke cigarettes / I stopped smoking cigarettes. (different meaning) 

There is already another thread about this change in meaning.

Most learners get confused by the infinitive to, thinking that after “to” the bare form of the verb must be followed. In fact, it's the opposite. A structure with looking forward to is always followed by a direct object (noun or gerund).

If you would think of a gerund as a noun, the following sentence may make more sense to you:

1. I am looking forward to my 2 week vacation in Hawaii.
   The direct object is the noun phrase “my 2 week vacation”
2. I am looking forward to spending two weeks off in Hawaii.
   The direct object is “spending 2 weeks”.

In # 2, "to spend 2 weeks off" is incorrect because it's not a direct object. 

Does this help?

The expression look forward to + ing (and others formed in the same way) often cause learners confusion, because, when studying gerunds and infinitives , they learn that the word to is usually followed by an infinitive:

• I want to go.
• I need to see you.
• I promise to give it to him.

Because of this, learners tend to write/say I look forward to see you, etc. This is incorrect.

The reason for this lies in the fact that the word to can be an infinitive marker (as in the three examples above), but it can also be a preposition. When to is a preposition, it can be followed either by a noun or by the -ing form of the verb:

• I look forward to our meeting.
• I look forward to meeting you.