  We obtain necessary and sufficient conditions for a supersymmetric field
configuration in the N=(1,0) U(1) or SU(2) gauged supergravities in six
dimensions, and impose the field equations on this general ansatz. It is found
that any supersymmetric solution is associated to an $SU(2)\ltimes
\mathbb{R}^4$ structure. The structure is characterized by a null Killing
vector which induces a natural 2+4 split of the six dimensional spacetime. A
suitable combination of the field equations implies that the scalar curvature
of the four dimensional Riemannian part, referred to as the base, obeys a
second order differential equation. Bosonic fluxes introduce torsion terms that
deform the $SU(2)\ltimes\mathbb{R}^4$ structure away from a covariantly
constant one. The most general structure can be classified in terms of its
intrinsic torsion. For a large class of solutions the gauge field strengths
admit a simple geometrical interpretation: in the U(1) theory the base is
K\"{a}hler, and the gauge field strength is the Ricci form; in the SU(2)
theory, the gauge field strengths are identified with the curvatures of the
left hand spin bundle of the base. We employ our general ansatz to construct
new supersymmetric solutions; we show that the U(1) theory admits a symmetric
Cahen-Wallach$_4\times S^2$ solution together with a compactifying pp-wave. The
SU(2) theory admits a black string, whose near horizon limit is $AdS_3\times
S_3$. We also obtain the Yang-Mills analogue of the Salam-Sezgin solution of
the U(1) theory, namely $R^{1,2}\times S^3$, where the $S^3$ is supported by a
sphaleron. Finally we obtain the additional constraints implied by enhanced
supersymmetry, and discuss Penrose limits in the theories.
