Urban systems have many heterogeneous sets and many heterogeneous relations between them. Network theory has been very successful in analysing binary relations and related dynamics and flows. Urban systems are also characterised by higher level relations between many elements, leading to a multidimensional generalisation of network theory. Many dynamic system properties are dependent on the underlying multidimensional connectivities. New relational mathematics can support a new theory of multilevel systems, in which levels are coherently integrated through lattice hierarchies. This multilevel multidimensional backcloth supports the traffic of human activity at all urban levels. Relational structure is transformed into numerical functions when moving from microto macro-levels. The relational backcloth includes the usual infrastructure of roads, buildings and land use activities. Their relatively slow structural dynamics contrasts with the relatively fast dynamics of human activity such as activity use and traffic flows. The backcloth has natural structural events defining system time in a way different to but intertwined with the clock time of physics. This method of representing urban systems allows planners to investigate interconnected multidimensional system trajectories of possible worlds, and to investigate how polices might play out within short and long-term time horizons. Also they support an approach to urban planning and design in which the system is always evolving and coevolving as it adapts to new and changing human requirements. The presentation assumes no prior knowledge and develops through examples illustrating the fundamental ideas.