Higher dimensional supersymmetry in 4D superspace

We present an explicit formulation of supersymmetric Yang-Mills theories from $\text{D}=$ 5 to 10 dimensions in the familiar $\mathbb{N}=1,\text{D}=4$ superspace. This provides the rules for globally supersymmetric model building with extra dimensions and in particular allows us to simply write down $\mathbb{N}=1$ SUSY preserving interactions between bulk fields and fields localized on branes. We present a few applications of the formalism by way of illustration, including supersymmetric ``shining’’ of bulk fields, orbifolds and localization of chiral fermions, anomaly inflow and super-Chern-Simons theories.