The occurrence of anomalies in the trace of the energy-momentum tensor for scalar field theories in curved space-time is discussed. For the special case of spherical space-time, an O(n+1)-covariant formalism is used to rederive the standard free-field anomaly in four dimensions, and to calculate the anomaly in six dimensions. It is then shown that for an interacting scalar field theory there is a further contribution to the trace anomaly proportional to the renormalization group β function. This assertion is then checked by explicit calculations in φ4 theory in four dimensions and φ3 theory in six dimensions and values for the anomaly found to fourth order in the renormalized coupling constants λ and g. Finally, these results are generalized to the case of an arbitrary background space-time, where it is shown that the introduction of a position-dependent coupling constant λ(x) enables the relation between the trace anomaly and the β function to be expressed in the form TμIμ=-β(λ)δWIδλ(x)|λ(x)=λ. where WI is the sum over vacuum bubble diagrams with interactions.