An internal combustion engine with a turbocharger, comprising: a turbocharger including a turbine disposed in an exhaust passage, the turbine being operative by exhaust energy of the internal combustion engine; an exhaust gas purifying catalyst, disposed in the exhaust passage at a downstream side of the turbine, for purifying an exhaust gas; an exhaust bypass passage for bypassing the turbine; a bypass valve for opening or closing the exhaust bypass passage; bypass valve opening control means for performing, if a fuel cut of the internal combustion engine is to be performed, during at least part of a period of time during which the fuel cut is performed, control to open the bypass valve or to make an opening angle of the bypass valve greater than an opening angle at a start of the fuel cut; and catalyst temperature acquiring means for detecting or estimating a temperature of the exhaust gas purifying catalyst, wherein the bypass valve opening control means performs the control to open the bypass valve or to make the opening angle of the bypass valve greater than the opening angle at the start of the fuel cut during the performance of the fuel cut and if the temperature of the exhaust gas purifying catalyst is higher than a first predetermined value; characterized in that: the bypass valve opening control means includes first returning-then-opening control performing means for performing the control to open the bypass valve or to make the opening angle of the bypass valve greater than the opening angle at the start of the fuel cut, if the temperature of the exhaust gas purifying catalyst is higher than a second predetermined value at the return from the fuel cut, the first returning-then-opening control performing means continuing performing the control until the temperature of the exhaust gas purifying catalyst is equal to, or less than, the second predetermined value, wherein the second predetermined value is lower than the first predetermined value. a turbocharger including a turbine disposed in an exhaust passage, the turbine being operative by exhaust energy of the internal combustion engine; an exhaust gas purifying catalyst, disposed in the exhaust passage at a downstream side of the turbine, for purifying an exhaust gas; an exhaust bypass passage for bypassing the turbine; a bypass valve for opening or closing the exhaust bypass passage; bypass valve opening control means for performing, if a fuel cut of the internal combustion engine is to be performed, during at least part of a period of time during which the fuel cut is performed, control to open the bypass valve or to make an opening angle of the bypass valve greater than an opening angle at a start of the fuel cut; and catalyst temperature acquiring means for detecting or estimating a temperature of the exhaust gas purifying catalyst, wherein the bypass valve opening control means performs the control to open the bypass valve or to make the opening angle of the bypass valve greater than the opening angle at the start of the fuel cut during the performance of the fuel cut and if the temperature of the exhaust gas purifying catalyst is higher than a first predetermined value; characterized in that: the bypass valve opening control means includes first returning-then-opening control performing means for performing the control to open the bypass valve or to make the opening angle of the bypass valve greater than the opening angle at the start of the fuel cut, if the temperature of the exhaust gas purifying catalyst is higher than a second predetermined value at the return from the fuel cut, the first returning-then-opening control performing means continuing performing the control until the temperature of the exhaust gas purifying catalyst is equal to, or less than, the second predetermined value, wherein the second predetermined value is lower than the first predetermined value. the bypass valve opening control means includes first returning-then-opening control performing means for performing the control to open the bypass valve or to make the opening angle of the bypass valve greater than the opening angle at the start of the fuel cut, if the temperature of the exhaust gas purifying catalyst is higher than a second predetermined value at the return from the fuel cut, the first returning-then-opening control performing means continuing performing the control until the temperature of the exhaust gas purifying catalyst is equal to, or less than, the second predetermined value, wherein the second predetermined value is lower than the first predetermined value.