The thermal conductivities kappa of the crystalline phases and amorphous solid states of water as well as clathrate hydrates are summarized and discussed. In particular, this review concerns the unusual temperature T and pressure p behaviors of kappa for some phases and states, which include glass-like K for crystalline clathrate hydrates and crystal-like kappa for low-density amorphous ice. The latter result implies that glassy water and low-density amorphous ice are different states. The results for the various phases and states are in most cases described well by the equations: kappa = D x T(-x) and 1n kappa = E + F x p, under isobaric and isothermal conditions, respectively. All phases that exhibit negative values for F are known to amorphize upon pressurization at low temperatures. Ice XI, which is obtained by annealing ice Ih below 70 K, exhibits positive F, which indicates that this phase does not amorphize like ice Ih upon pressurization.