Learn French: Le contact arrive en dépit de la surveillance. - The contact arrives in spite of the surveillance.

Both mean "in spite of" or "despite," and both must be followed directly by a noun or pronoun. En dépit de is simply a bit more formal and elevated than malgré. You could easily swap it in this sentence to say malgré la surveillance with no change in meaning.

What happens to en dépit de if the noun after it is masculine or plural?

The de in en dépit de follows the normal rules of contraction. Because surveillance is feminine, we use de la. If it were a masculine noun like le danger, it would contract to en dépit du danger. For a plural noun like les caméras, it would become en dépit des caméras.

Can I follow en dépit de with a conjugated verb, like "en dépit d'ils surveillent"?

No, en dépit de is a prepositional phrase and must always be followed by a noun or pronoun, never a conjugated verb. If you want to use a clause with a verb (meaning "despite the fact that they are watching"), you would need a conjunction like bien que followed by a subject and a subjunctive verb.

Does arrive only mean physical arrival here?