Almost all the numerous forms of ice including 13 crystalline phases, several distinct amorphous states as well as clathrate hydrates have been characterized in terms of their thermal conductivity κ. A few deviations from both the typically strongly decreasing κ(T) (κ～T^-1) associated with crystals and the weakly increasing κ(T) associated with glassy states are observed. In particular, the crystalline clathrate hydrates show glass-like k, whereas low-density amorphous (LDA) ice shows crystal-like κ(κ～T^-0.6). The latter is unique for an amorphous state and indicates that LDA ice exhibits a surprisingly high degree of structural order that allows for high-frequency phonon propagation. It also implies that LDA ice is not the glassy counterpart of ambient liquid water that shows typical amorphous-like behavior. Moreover, the crystalline ices Ih and Ic and LDA ice exhibit abnormal negative pressure coefficients of κ. Both ices Ih and Ic demonstrate pressure-induced amorphization to high-density amorphous (HDA) ice, a state to which also LDA ice transforms on pressurization. The negative pressure coefficient is a signature of phonon softening which leads to a disordering transition upon pressurization.