  We describe T-duality between general geometric and non-geometric backgrounds
as higher groupoid bundles with connections. Our description extends the
previous observation by Nikolaus and Waldorf that the topological aspects of
geometric and half-geometric T-dualities can be described in terms of higher
geometry. We extend their construction in two ways. First, we endow the higher
geometries with adjusted connections, which allow us to discuss explicit
formulas for the metric and the Kalb-Ramond field of a T-background. Second, we
extend the principal 2-bundles to augmented 2-groupoid bundles, which
accommodate the scalar fields arising in T-duality along several directions as
well as $Q$- and $R$-fluxes. Our description is manifestly covariant under the
full T-duality group $\mathsf{GO}(n,n;\mathbb{Z})$, and it has interesting
physical and mathematical implications.
