The celebrated hook-length formula of Frame, Robinson and Thrall from 1954 gives a product formula for the number of standard Young tableaux of straight shape. No such product formula exists for skew shapes. In 2014, Naruse announced a formula for skew shapes as a positive sum of products of hook-lengths using "excited diagrams" [ Ikeda-Naruse, Kreiman, Knutson-Miller-Yong]. We prove NaruseA's formula algebraically and combinatorially, and we exhibit a bijection between SSYTs or reverse plane partitions of skew shape and certain integer arrays. We also give two q-analogues of this formula involving semistandard tableaux and reverse plane partitions of skew shape. We end by applying our results to border strips to obtain relations between Euler numbers and Dyck paths, and uncover some unusual identities. Joint work with Alejandro Morales and Igor Pak.