  We introduce graphical complexes of groups, which can be thought of as a
generalisation of Coxeter systems with 1-dimensional nerves. We show that these
complexes are strictly developable, and we equip the resulting Basic
Construction with three structures of non-positive curvature: piecewise linear
CAT(0), C(6) graphical small cancellation, and a systolic one. We then use
these structures to establish various properties of the fundamental groups of
these complexes, such as biautomaticity and Tits Alternative. We isolate an
easily checkable condition implying hyperbolicity of the fundamental groups,
and we construct some non-hyperbolic examples. We also briefly discuss a
parallel theory of C(4)-T(4) graphical complexes of groups and outline their
basic properties.
