Let X$$X$$ be a smooth projective variety of dimension n$$n$$ over C$$\mathbb {C}$$ equipped with a very ample Hermitian line bundle L$$\mathcal {L}$$. In the first part of the paper, we show that if there exists a toric degeneration of X$$X$$ satisfying some natural hypotheses (which are satisfied in many settings), then there exists a surjective continuous map from X$$X$$ to the special fiber X0$$X_0$$ which is a symplectomorphism on an open dense subset U$$U$$. From this we are then able to construct a completely integrable system on X$$X$$ in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions {H1,…,Hn}$$\{H_1, \ldots , H_n\}$$ on X$$X$$ which are continuous on all of X$$X$$, smooth on an open dense subset U$$U$$ of X$$X$$, and pairwise Poisson-commute on U$$U$$. Moreover, our integrable system in fact generates a Hamiltonian torus action on U$$U$$. In the second part, we show that the toric degenerations arising in the theory of Newton-Okounkov bodies satisfy all the hypotheses of the first part of the paper. In this case the image of the ‘moment map’ μ=(H1,…,Hn):X→Rn$$\mu = (H_1, \ldots , H_n): X \rightarrow \mathbb {R}^n$$ is precisely the Newton-Okounkov bodyΔ=Δ(R,v)$$\Delta = \Delta (R, v)$$ associated to the homogeneous coordinate ring R$$R$$ of X$$X$$, and an appropriate choice of a valuation v$$v$$ on R$$R$$. Our main technical tools come from algebraic geometry, differential (Kähler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Łojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties X$$X$$, this manuscript provides a rich source of new examples of integrable systems. We discuss concrete examples, including elliptic curves, flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties.