  The celebrated hook-length formula gives a product formula for the number of
standard Young tableaux of a straight shape. In 2014, Naruse announced a more
general formula for the number of standard Young tableaux of skew shapes as a
positive sum over excited diagrams of products of hook-lengths. We give an
algebraic and a combinatorial proof of Naruse's formula, by using factorial
Schur functions and a generalization of the Hillman--Grassl correspondence,
respectively.
  The main new results are two different $q$-analogues of Naruse's formula: for
the skew Schur functions, and for counting reverse plane partitions of skew
shapes. We establish explicit bijections between these objects and families of
integer arrays with certain nonzero entries, which also proves the second
formula.
