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We provide a complete classification of the class of unital graph$C^*$-algebras - prominently containing the full family of Cuntz-Kriegeralgebras - showing that Morita equivalence in this case is determined byordered, filtered $K$-theory. The classification result is geometric in thesense that it establishes that any Morita equivalence between $C^*(E)$ and$C^*(F)$ in this class can be realized by a sequence of moves leading from $E$to $F$, in a way resembling the role of Reidemeister moves on knots. As a keyingredient, we introduce a new class of such moves, establish that they leavethe graph algebras invariant, and prove that after this augmentation, the listof moves becomes complete in the sense described above.  Along the way, we prove that every ordered, reduced filtered $K$-theoryisomorphism can be lifted to an isomorphism between the stabilized$C^*$-algebras - and, as a consequence, that every ordered, reduced filtered$K$-theory isomorphism preserving the class of the unit comes from a$*$-isomorphism between the unital graph $C^*$-algebras themselves.  It follows that the question of Morita equivalence amongst unital graph$C^*$-algebras is a decidable one. As immediate examples of applications of ourresults we revisit the classification problem for quantum lens spaces andverify, in the unital case, the Abrams-Tomforde conjectures.

The complete classification of unital graph $C^*$-algebras: Geometric and strong