The article presents the history and current problems of applying mineral tailings slurry deep thickening to paste condition technology, and its disposal at mining-and-concentrating facilities. It is noted, that this technology application currently becomes more and more vital owing to increasing necessity of providing safety for hydraulic-engineering systems and structures, as well as saving energy and water, and for environmental reasons. It is shown that justification of economic advisability of tailings slurry deep thickening and paste disposal should be based on existing operating parameters and conditions of a particular processing facility, with consideration of its development prospects. This point is explained by the example of two Chilean concentrating plants — copper ore Codelco Norte (DCN) and the Serro Negro iron ore deposit.