  We construct non-critical pure spinor superstrings in two, four and six
dimensions. We find explicitly the map between the RNS variables and the pure
spinor ones in the linear dilaton background. The RNS variables map onto a
patch of the pure spinor space and the holomorphic top form on the pure spinor
space is an essential ingredient of the mapping. A basic feature of the map is
the requirement of doubling the superspace, which we analyze in detail. We
study the structure of the non-critical pure spinor space, which is different
from the ten-dimensional one, and its quantum anomalies. We compute the pure
spinor lowest lying BRST cohomology and find an agreement with the RNS spectra.
The analysis is generalized to curved backgrounds and we construct as an
example the non-critical pure spinor type IIA superstring on AdS_4 with RR
4-form flux.
