The paper discusses the cosmological constant problem in the context of dilatation symmetry and its possible anomalies. For dilatation symmetric quantum theories, realistic asymptotic cosmology is achievable if the effective potential has a non-trivial minimum. However, for theories with a dilatation anomaly, a non-trivial "cosmon condition" is required, where the energy-momentum tensor in the vacuum is purely anomalous. This condition is related to the short-distance renormalization group behavior of the fundamental theory. Observable deviations from the standard hot big bang cosmology are possible. The author first investigates models without intrinsic mass scales, where the effective potential depends on the ratio of fields rather than their absolute values. In such models, the cosmology can be consistent with the standard big bang picture if the potential has a non-trivial minimum. For models with dilatation anomalies, the author formulates three conditions for a realistic cosmology: the trace anomaly should vanish at some value of the dilaton field, the dilaton mass should be positive, and the trace of the energy-momentum tensor in the static vacuum solution should be purely anomalous. If these conditions are met, the dilaton is called a "cosmon," and its dynamics drives the cosmological constant to zero. The paper also explores the connection between the "cosmon condition" and the short-distance behavior of the underlying fundamental theory, particularly in models where intrinsic mass scales arise from the running of dimensionless couplings. The author discusses a renormalization group equation for the cosmological "constant" and its implications for the cosmology. Finally, the paper concludes with a discussion of the cosmological constant problem, emphasizing that realistic cosmology can be achieved if the anomalous dimension of the effective potential is within a certain range.The paper discusses the cosmological constant problem in the context of dilatation symmetry and its possible anomalies. For dilatation symmetric quantum theories, realistic asymptotic cosmology is achievable if the effective potential has a non-trivial minimum. However, for theories with a dilatation anomaly, a non-trivial "cosmon condition" is required, where the energy-momentum tensor in the vacuum is purely anomalous. This condition is related to the short-distance renormalization group behavior of the fundamental theory. Observable deviations from the standard hot big bang cosmology are possible. The author first investigates models without intrinsic mass scales, where the effective potential depends on the ratio of fields rather than their absolute values. In such models, the cosmology can be consistent with the standard big bang picture if the potential has a non-trivial minimum. For models with dilatation anomalies, the author formulates three conditions for a realistic cosmology: the trace anomaly should vanish at some value of the dilaton field, the dilaton mass should be positive, and the trace of the energy-momentum tensor in the static vacuum solution should be purely anomalous. If these conditions are met, the dilaton is called a "cosmon," and its dynamics drives the cosmological constant to zero. The paper also explores the connection between the "cosmon condition" and the short-distance behavior of the underlying fundamental theory, particularly in models where intrinsic mass scales arise from the running of dimensionless couplings. The author discusses a renormalization group equation for the cosmological "constant" and its implications for the cosmology. Finally, the paper concludes with a discussion of the cosmological constant problem, emphasizing that realistic cosmology can be achieved if the anomalous dimension of the effective potential is within a certain range.