Higher dimensional supersymmetry in 4D superspace

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