We study the general class of gravitational field theories constructed on the basis of scale invariance (and therefore absence of any mass parameters) and invariance under transverse diffeomorphisms (TDiff), which are the 4-volume conserving coordinate transformations. We show that these theories are equivalent to a specific type of scalar-tensor theories of gravity (invariant under all diffeomorphisms) with a number of properties, making them phenomenologically interesting. In particular, they lead to the evolution of the universe supported by present observations: inflation in the past, followed by the radiation and matter dominated stages and accelerated expansion at present. All mass scales in this type of theories come from one and the same source. The massless particle spectrum of these theories contains the graviton and a new particle -- dilaton, which has only derivative couplings and thus escapes the fifth force constraints.