Symmetries of a classical action functional describing a physical system (conservation laws for such symmetries were investigated by Emmy Noether), can "break" after a quantization procedure when going to the quantized action functional. Anomalies in physics measure this "breaking" of symmetry at the quantum level. We want to relate this type of anomaly to anomalies one encounters in mathematics when extracting finite parts of diverging quantities. For example, the zeta-determinant, which is an essential tool to define quantized actions functional in physics, is a way of extracting a finite part from an otherwise diverging determinant, the eta-invariant is a way to define the phase of such a zeta-determinant. Some anomalies in physics lie in logarithmic variations of partition functions defined by zeta-regularization techniques, and can be interpreted as regularized traces of some pseudo-differential operator. For the sake of simplicity (even if this is rather non physical since it boils down to working in a 0 dimensional space !), we shall focus on regularized traces of operators on the circle, describing tracial anomalies accuring from taking finite parts. We relate local terms arising in index theory and anomalies in quantum field theory to such tracial anomalies.