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In this paper we prove the existence of extremal functions for theAdams-Moser-Trudinger inequality on the Sobolev space $H^{m}(\Omega)$, where$\Omega$ is any bounded, smooth, open subset of $\mathbb{R}^{2m}$, $m\ge 1$.Moreover, we extend this result to improved versions of Adams' inequality ofAdimurthi-Druet type. Our strategy is based on blow-up analysis for sequencesof subcritical extremals and introduces several new techniques andconstructions. The most important one is a new procedure for obtainingcapacity-type estimates on annular regions.

Improved Adams-type inequalities and their extremals in dimension 2m