[verb (plain)] + にもかかわらず [i-adj] + にもかかわらず [na-adj] + な / である + にもかかわらず [noun] + にもかかわらず [noun] + である + にもかかわらず
にもかかわらず means "despite", "even though": the second half of the sentence is not what the first half would lead you to expect. It goes after the plain form of a verb or an い-adjective. A な-adjective takes な or である in front of it, and a noun takes it directly or with である – never with だ.
It is usually written in kana. The kanji spelling にもかかわら関わらず is common too, and so is the older にもかかわら拘らず.
Like のに, it reports what actually happened. The second half cannot be a request, an invitation or what you intend to do.
Without も, にかかわらず is a different pattern: "regardless of", "whether or not". It follows a noun that covers a range – age, weather, experience – and says the outcome does not depend on it.
On its own at the head of a sentence, にもかかわらず means "nevertheless" and points back to the sentence before.