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We obtain necessary and sufficient conditions for a supersymmetric fieldconfiguration in the N=(1,0) U(1) or SU(2) gauged supergravities in sixdimensions, and impose the field equations on this general ansatz. It is foundthat any supersymmetric solution is associated to an $SU(2)\ltimes\mathbb{R}^4$ structure. The structure is characterized by a null Killingvector which induces a natural 2+4 split of the six dimensional spacetime. Asuitable combination of the field equations implies that the scalar curvatureof the four dimensional Riemannian part, referred to as the base, obeys asecond order differential equation. Bosonic fluxes introduce torsion terms thatdeform the $SU(2)\ltimes\mathbb{R}^4$ structure away from a covariantlyconstant one. The most general structure can be classified in terms of itsintrinsic torsion. For a large class of solutions the gauge field strengthsadmit a simple geometrical interpretation: in the U(1) theory the base isK\"{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 theleft hand spin bundle of the base. We employ our general ansatz to constructnew supersymmetric solutions; we show that the U(1) theory admits a symmetricCahen-Wallach$_4\times S^2$ solution together with a compactifying pp-wave. TheSU(2) theory admits a black string, whose near horizon limit is $AdS_3\timesS_3$. We also obtain the Yang-Mills analogue of the Salam-Sezgin solution ofthe U(1) theory, namely $R^{1,2}\times S^3$, where the $S^3$ is supported by asphaleron. Finally we obtain the additional constraints implied by enhancedsupersymmetry, and discuss Penrose limits in the theories.Comment: 1+29 pages, late

The general form of supersymmetric solutions of N = (1, 0) U (1) and SU (2) gauged supergravities in six dimensions