An action of a torus $T$ on a manifold $M$ is locally standard if, at each point, the stabilizer is a sub-torus and the non-zero isotropy weights are a basis to its weight lattice.

The quotient $M/T$ is then a manifold-with-corners, equipped with so called a unimodular labelling, which keeps track of the isotropy representations in $M$ (and which is equivalent to the characteristic function if $M/T$ is a simple convex polytope).

The twistedness of $M$ over $M/T$ is encoded by a degree two cohomology class on $M/T$ with coefficients in the integral lattice $\mathfrak{t}_{\mathbb{Z}}$ of the Lie algebra of $T$.

In this talk, we give a classification of locally standard smooth actions of $T$, up to equivariant diffeomorphism, in terms of triples $(Q, \lambda, c)$, where $Q$ is a manifold-with-corners, $\lambda$ is a unimodular labelling, and $c$ is a degree two cohomology class with coefficients in $\mathfrak{t}_{\mathbb{Z}}$.

This is a joint work with Yael Karshon.