  In this paper, we formally introduce the notion of Ap{\'e}ry-like sums and we
show that every multiple zeta values can be expressed as a $\bf Z$-linear
combination of them. We even describe a canonical way to do so. This allows us
to put in a new theoretical context several identities scattered in the
literature, as well as to discover many new interesting ones. We give in this
paper new integral formulas for multiple zeta values and Ap\'ery-like sums.
They enable us to give a short direct proof of Zagier's formulas for
$\zeta(2,\ldots,2,3,2,\ldots,2)$ as well as of similar ones in the context of
Ap{\'e}ry-like sums. The relations between Ap\'ery-like sums themselves still
remain rather mysterious, but we get significant results and state some
conjectures about their pattern.
