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Materials with memory enable to model the propagation of heat by finite speed. Models with memory are also used in phase transition problems and thermoelasticity. In the present book inverse problems to determine kernels of heat flux and internal energy in the one- dimensional non-homogeneous heat flow are studied. We consider the case when these kernels are degenerate, i.e. representable as sums of the known space-dependent functions times the unknown time- dependent coefficients. We proved existence and uniqueness of two problems of such kind. The first one is a problem with purely temperature observations. Then the kernel of internal energy is determined with higher smoothness than the kernel of heat flux. The second one is te problem with purely flux observations. Then the kernels of internal energy and heat flux are determined with the same level of accuracy. Moreover, it has been shown that the homogeneous inverse problem with flux observations is severely il-posed. The method involves application of the Laplace transform and reduction of the transformed problem to a fixed-point form.