A method for applying and curing an organic polymer coating onto a substrate comprising the steps of: 1) under non-polymerizing conditions, applying a curable coating composition that at 22°C and at atmospheric pressure is a liquid or a suspension of one or more solids in a liquid phase to at least one surface of the substrate, wherein the curable coating composition contains i) at least one polymerizable monomer that polymerizes in the presence of free radicals, has exactly one free-radical-polymerizable group per molecule and has a boiling temperature of at least 100°C at one atmosphere pressure, and ii) at least one heat-activated polymerization initiator, which curable composition contains no more than 10% by weight of organic compounds that have boiling temperatures below 100°C at one atmosphere pressure and no more than 5% by weight water, based on the entire weight of the coating composition; then 2) under non-polymerizing conditions, purging molecular oxygen from the interior of a vessel containing the substrate with the applied curable coating, then 3) during and/or after step 2), pressurizing the purged vessel containing the substrate with the applied curable coating with an oxygen-deficient gas to a gas pressure of at least 120 kPa actual, and then, without exposing the substrate with applied curable coating to an atmosphere containing 1 mole-percent or more oxygen, 4) at least partially curing the curable coating composition by heating the substrate with the applied curable coating composition under oxygen-deficient conditions to a temperature sufficient to activate the heat-activated polymerization initiator and initiate polymerization of the at least one polymerizable monomer, to form an at least partially cured organic polymer coating on the substrate. 1) under non-polymerizing conditions, applying a curable coating composition that at 22°C and at atmospheric pressure is a liquid or a suspension of one or more solids in a liquid phase to at least one surface of the substrate, wherein the curable coating composition contains i) at least one polymerizable monomer that polymerizes in the presence of free radicals, has exactly one free-radical-polymerizable group per molecule and has a boiling temperature of at least 100°C at one atmosphere pressure, and ii) at least one heat-activated polymerization initiator, which curable composition contains no more than 10% by weight of organic compounds that have boiling temperatures below 100°C at one atmosphere pressure and no more than 5% by weight water, based on the entire weight of the coating composition; then 2) under non-polymerizing conditions, purging molecular oxygen from the interior of a vessel containing the substrate with the applied curable coating, then 3) during and/or after step 2), pressurizing the purged vessel containing the substrate with the applied curable coating with an oxygen-deficient gas to a gas pressure of at least 120 kPa actual, and then, without exposing the substrate with applied curable coating to an atmosphere containing 1 mole-percent or more oxygen, 4) at least partially curing the curable coating composition by heating the substrate with the applied curable coating composition under oxygen-deficient conditions to a temperature sufficient to activate the heat-activated polymerization initiator and initiate polymerization of the at least one polymerizable monomer, to form an at least partially cured organic polymer coating on the substrate.