Higher dimensional supersymmetry in 4D superspace

We present an explicit formulation of supersymmetric Yang-Mills theories from D = 5 to 10 dimensions in the familiar N = 1, D = 4 superspace. This provides the rules for globally supersymmetric model building with extra dimensions and in particular allows us to simply write down 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.