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This is a review on the two first parts of our work on $p$-adic multiple zetavalues at $N$-th roots of unity ($p$MZV$\mu_{N}$'s), the $p$-adic periods ofthe crystalline pro-unipotent fundamental groupoid of $\mathbb{P}^{1} -\{0,\mu_{N},\infty\}$ (where $N$ and $p$ are coprime). We restrict forsimplicity the review to the case of $N=1$, i.e. the case of $p$-adic multiplezeta values ($p$MZV's). The main tools are new objects which we call $p$-adicpro-unipotent harmonic actions. These are continuous group actions on a spacecontaining the non-commutative generating series of weighted multiple harmonicsums, they are related to the motivic Galois action on$\pi_{1}^{\un}(\mathbb{P}^{1} - \{0,1,\infty\})$ and to the Poisson-Iharabracket, and interrelated by some maps. They are defined in \cite{J2} and\cite{J3} ; the definition relies on a simplification of the differentialequation of the Frobenius, proved as a preliminary technical fact by \cite{J1}.Part I (\cite{J1},\cite{J2},\cite{J3}) is an explicit computation of theFrobenius of $\pi_{1}^{\un,\crys}(\mathbb{P}^{1} - \{0,1,\infty\})$, and inparticular of $p$MZV's. We give formulas which keep a track of the motivicGalois action. Part II (\cite{J4},\cite{J5},\cite{J6}) is a study of thealgebraic properties of $p$MZV's brought together with the formulas of part I.We state an explicit elementary version of the Galois theory of $p$MZV's.

$p$-adic multiple zeta values and $p$-adic pro-unipotent harmonic actions : summary of parts I and II