  The cyclotomic $p$-adic multi-zeta values are the $p$-adic periods of
$\pi_{1}(\mathbb{G}_{m} \setminus \mu_{M},\cdot),$ the unipotent fundamental
group of the multiplicative group minus the $M$-th roots of unity. In this
paper, we compute the cyclotomic $p$-adic multi-zeta values at all depths. This
paper generalizes the results in [6] and [7]. Since the main result gives
explicit formulas we expect it to be useful in proving non-vanishing and
transcendence results for these $p$-adic periods and also, through the use of
$p$-adic Hodge theory, in proving non-triviality results for the corresponding
$p$-adic Galois representations.
