A recent trend in inverse scattering theory has focused on the development of a qualitative approach, which yield fast reconstructions with very little of a priori information  but at the expense of obtaining only limited information of the scatterer such as the  support, and estimates on the values of the constitutive parameters. Examples of such an approach are the linear sampling and factorization methods. These two methods are very well developed  in the time harmonic regime,  more generally for the underlying elliptic PDEs models, however for hyperbolic problems only limited results are available. In inverse scattering, the use  of  time domain  measurements  is  a remedy  for  large  amount  of  spatial  data  typically  needed  for the  application  of  qualitative  approach.

In this presentation we will discuss recent progress  in the development of linear sampling and factorization methods in the time domain.  Fist we consider the linear sampling method for solving inverse scattering problem for inhomogeneous media.  A   fundamental   tool   for  the justification of this method  is  the  solvability  of  the time  domain   interior   transmission   problem that relies on understanding the location on the complex plane of transmission eigenvalues. We present some latest results in this regard. The second problem addresses  the lack of mathematical rigorousness of the linear sampling method. In this context we discuss the factorization method to obtain explicit characterization of a (possibly non-convex) Dirichlet  scattering object from measurements of time-dependent causal scattered waves in the far field regime. In particular, we prove that far fields of  solutions to the wave equation due to particularly  modified  incident waves, characterize the obstacle  by a range criterion involving the square root of the time derivative  of the corresponding  far field operator. Our analysis  makes essential use of a coercivity property of the solution of the Dirichlet initial boundary value problem for the wave equation in the Laplace domain that forces us to consider this  particular modification of the far field operator. The latter in fact, can be chosen arbitrarily close to the true far field operator given in terms of physical measurements. Finally we discuss some related open questions.