I am a US citizen living abroad in Europe and working for a New Jersey company which has no office in Europe.
What tax return I have to file for Federal Income tax and what are my NJ tax obligations?

Redundant question; see below.

You need to file Form 1040. If you are paying taxes to a European country, you can mitigate your U.S. federal tax obligation by claiming the Foreign Tax Credit (Form 1116) or the Foreign Earned Income Exclusion (Form 2555), or a combination of the two.

If you maintain a home in Jew Jersey, or intend to return there to live, you must pay NJ income tax on your world-wide income.

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Originally Posted by AtlantaTaxExpert

If you maintain a home in Jew Jersey, or intend to return there to live, you must pay NJ income tax on your world-wide income.

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Just to add, given that the OP made no mention of having a NJ home - if NJ is not his legal residence then he does not need to file NJ income tax unless he has NJ-source income. Being paid by a company that happens to headquartered in NJ does not mean his pay is NJ-source, so it all likelihood no NK tax filing is required.

For any other atate, I agree, bur NJ takes the position that you are a NJ resident while overseas if you plan to return to NJ after the overseas tour is over, and they ASSUME you are coming back if you leave from NJ.