Quantum field theories in higher dimensions

Weakly coupled interacting quantum field theories exist only in four or less space-time dimensions. However, there are arguments that consistent quantum field theories exist also in five and six space-time dimensions. In particular there are indirect arguments for the existence of local superconformal field theories in five and six dimensions, and of non-local field theories (called "little string theories") in six dimensions. Can we provide direct constructions of these higher dimensional theories, and understand their properties ? What can we learn about lower dimensional field theories by compactifying these theories on various manifolds ? Are there any consistent field theories above six space-time dimensions, and are there any consistent non-supersymmetric field theories above four space-time dimensions ? What are the rules for dealing with non-local (but non-gravitational) theories like "little string theories" ?