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We discuss non-Lorentzian Lagrangian field theories in 2n − 1 dimensions that admit an SU(1, n) spacetime symmetry which includes a scaling transformation. These can be obtained by a conformal compactification of a 2n-dimensional Minkowskian conformal field theory. We discuss the symmetry algebra, its representations including primary fields and unitarity bounds. We also give various examples of free theories in a variety of dimensions and a discussion of how to reconstruct the parent 2n-dimensional theory.