The construction of a quantum field theory amounts to finding an RG trajectory which is infinitely extended in the sense that the action functional described by { g α α ( k ) } is well-behaved for all values of the momentum scale parameter k , including the infrared limit k → → 0 and the ultraviolet (UV) limit k → → ∞ ∞ . Asymptotic safety is a way of dealing with the latter limit. Its fundamental requirement is the existence of a fixed point of the RG flow. By definition this is a point { g α α ∗ ∗ } in the theory space where the running of all couplings stops, or, in other words, a zero of all beta-functions : β β γ γ ( { g α α ∗ ∗ } ) = 0 for all γ γ . In addition that fixed point must have at least one UV-attractive direction. This ensures that there are one or more RG trajectories which run into the fixed point for increasing scale. The set of all points in the theory space that are "pulled" into the UV fixed point by going to larger scales is referred to as UV critical surface. Thus the UV critical surface consists of all those trajectories which are safe from UV divergences in the sense that all couplings approach finite fixed point values as k → → ∞ ∞ . The key hypothesis underlying asymptotic safety is that only trajectories running entirely within the UV critical surface of an appropriate fixed point can be infinitely extended and thus define a fundamental quantum field theory. It is obvious that such trajectories are well-behaved in the UV limit as the existence of a fixed point allows them to "stay at a point" for an infinitely long RG "time".