Consider a convex domain on the plane and the associated

billiard inside. The length spectrum is the closure of the union of

perimeters of all period orbits. The length spectrum is closely

related to the Laplace spectrum, through the wave trace and

the well-known question: ``Can you hear the shape of a drum?''

A domain is called dynamically spectrally rigid if any smooth

deformation preserving the length spectrum is an isometry.

During the talk I will discuss recent results on dynamical

spectral rigidity of convex planar domains.