In this work, we aim to establish the main regularities of hydrothermal precipitation of copper from the previously unstudied sulfuric-nitric acid and sulfate solutions of atmospheric and autoclave processing of metallurgical dusts. Pyrite was used as a sulfidizer. The elemental composition of the products was determined by X-ray fluorescence, inductively coupled plasma atomic emission spectrometry, and atomic absorption analysis. It was found that copper precipitation by the proposed technology enables its recovery at a level of more than 95%. For sulfate solutions with a high arsenic concentration, a two-stage scheme of oxidation-hydrothermal treatment with the following parameters was proposed: temperature 180°С, duration 2 h, oxygen consumption 0.026 dm3/g pyrite (for the 1st stage), and 200°С and 2 h (for the 2nd stage). Extraction in the cake amounted to 95.4% of copper (in the form of Cu2S) and 91.4% of arsenic (in the form of FeAsO4), which allowed further separation of these metals by flotation. An autoclave treatment of a Cu–Zn–Fe–As–NO3 solution in the presence of pyrite at 180–220°С produced the activation energy values (kJ/mol) corresponding to the kinetic regime calculated by the two methods: 64.6 by the conventional method and 60.5 by the shrinking core model. The kinetic parameters of CuSO4–H2SO4–FeS2–H2O and CuNO3–H2SO4–FeS2–H2O systems were also determined. Flotation enrichment of copper autoclave precipitation cake was shown to result in a high degree of Cu and As separation, with the recovery amounting to Cu ˃ 95% and As ˂ 5%. Precious metals contained in pyrite are transferred to a flotation concentrate by 92.7% (Au) and 96.5% (Ag). The composition of the resulting flotation concentrate comprised (%): 12 Cu; 37 Fe; 38 S; 13 other elements. It is shown that, in order to obtain a product with the required content of copper, flotation concentrate should be separated into pyrite and copper concentrates followed by an additional flotation of primary copper concentrate in an alkaline medium in the presence of lime. Hence, our study has established the main regularities of hydrothermal precipitation of copper from the sulfuric-nitric acid and sulfate solutions of atmospheric and autoclave processing of metallurgical dusts.