  We present new classes of explicit supersymmetric AdS_3 solutions of type IIB
supergravity with non-vanishing five-form flux and AdS_2 solutions of D=11
supergravity with electric four-form flux. The former are dual to
two-dimensional SCFTs with (0,2) supersymmetry and the latter to supersymmetric
quantum mechanics with two supercharges. We also investigate more general
classes of AdS_3 solutions of type IIB supergravity and AdS_2 solutions of D=11
supergravity which in addition have non-vanishing three-form flux and magnetic
four-form flux, respectively. The construction of these more general solutions
makes essential use of the Chern-Simons or "transgression" terms in the Bianchi
identity or the equation of motion of the field strengths in the supergravity
theories. We construct infinite new classes of explicit examples and for some
of the type IIB solutions determine the central charge of the dual SCFTs. The
type IIB solutions with non-vanishing three-form flux that we construct include
a two-torus, and after two T-dualities and an S-duality, we obtain new AdS_3
solutions with only the NS fields being non-trivial.
