  Attempts to solve naturalness by having the weak scale as the only breaking
of classical scale invariance have to deal with two severe difficulties:
gravity and the absence of Landau poles. We show that solutions to the first
problem require premature modifications of gravity at scales no larger than
$10^{11}$ GeV, while the second problem calls for many new particles at the
weak scale. To build models that fulfil these properties, we classify
4-dimensional Quantum Field Theories that satisfy Total Asymptotic Freedom
(TAF): the theory holds up to infinite energy, where all coupling constants
flow to zero. We develop a technique to identify such theories and determine
their low-energy predictions. Since the Standard Model turns out to be
asymptotically free only under the unphysical conditions $g_1 = 0$, $M_t = 186$
GeV, $M_\tau = 0$, $M_h = 163$ GeV, we explore some of its weak-scale
extensions that satisfy the requirements for TAF.
