  We discuss chiral zero-rest-mass field equations on six-dimensional
space-time from a twistorial point of view. Specifically, we present a detailed
cohomological analysis, develop both Penrose and Penrose-Ward transforms, and
analyse the corresponding contour integral formulae. We also give twistor space
action principles. We then dimensionally reduce the twistor space of
six-dimensional space-time to obtain twistor formulations of various theories
in lower dimensions. Besides well-known twistor spaces, we also find a novel
twistor space amongst these reductions, which turns out to be suitable for a
twistorial description of self-dual strings. For these reduced twistor spaces,
we explain the Penrose and Penrose-Ward transforms as well as contour integral
formulae.
