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We show that there exist supersymmetric Lifshitz vacua in off-shellEinstein-Weyl supergravity, in addition to the BPS AdS_4 vacuum. The Lifshitzexponents are determined by the product of the cosmological constant and thecoupling of the Weyl-squared term. We then obtain the equations of thesupersymmetric solutions that are asymptotic to the AdS or Lifshitz vacua. Weobtain many examples of exact solutions as well as numerical ones. We findexamples of extremal AdS black branes whose near-horizon geometry isAdS_2\times T^2. We also find an extremal Lifshitz black hole with z=-2, whosehorizon coincides with the curvature singularity. However the asymptoticLifshitz solutions are in general smooth wormholes. In conformal supergravity,we find intriguing examples of non-extremal "BPS" AdS and Lifshitz black holeswhose local Killing spinor is divergent on the horizon. We show that all thesupersymmetric asymptotic AdS and Lifshitz solutions have the vanishing Noethercharge associated with some scaling symmetry. We also study the integrabilitycondition of the Killing spinor equation and the supersymmetric invariance ofthe action. Finally we show that the only spherically-symmetric BPS solution isthe AdS vacuum.

Supersymmetric Asymptotic AdS and Lifshitz Solutions in Einstein-Weyl and Conformal Supergravities