As an alternative to the Big Bang (the standard model), we present a mathematical theory of cosmological redshift. We show that a fundamental formula of Lobachevskian (hyperbolic) geometry describes cosmological redshift and the Doppler effect as well. As presented here, the cosmological redshift preserves wavelength ratios (it shifts uniformly the whole electromagnetic spectrum), it is scale invariant, it is a monotonically increasing function of distance, and it is source independent. It agrees with all experimental data. The distortion introduced by imaging from hyperbolic into Euclidean space and the limitations of Special Relativity are discussed. Physical observations in Lobachevskian space are discussed and the new formula relating redshift and/or Doppler shift to aberration is given. An analysis is presented of an erroneous origin of Hubble's so called velocity distance law.

PACS numbers: 98.70.Vc, 04.80.Nn, 98.80.–k, 98.80.Es