An exhaust purifying apparatus for an internal combustion engine, the apparatus comprising: an upstream catalyst disposed in an exhaust path of the internal combustion engine, the upstream catalyst storing a predetermined component of exhaust gas in an oxidizing atmosphere and releasing the predetermined component in a reducing atmosphere; a downstream catalyst disposed downstream of the upstream catalyst in the exhaust path, the downstream catalyst storing the predetermined component of the exhaust gas in an oxidizing atmosphere and releasing the predetermined component in a reducing atmosphere; and characterised by a controller adapted to execute a first purging control if a quantity of the predetermined component stored in the upstream catalyst becomes equal to or larger than a first threshold, the first purging control involving release of the predetermined component from the upstream catalyst through supply of a reductant, and to execute a second purging control if a quantity of the predetermined component stored in the downstream catalyst becomes equal to or larger than a second threshold, the second purging control involving release of the predetermined component from the downstream catalyst through supply of the reductant, wherein if the quantity of the predetermined component stored in the upstream catalyst exceeds a third threshold and if the quantity of the predetermined component stored in the downstream catalyst exceeds a fourth threshold, the controller simultaneously and continuously executes the first purging control and the second purging control, the third threshold being smaller than the first threshold, the fourth threshold being smaller than the second threshold. an upstream catalyst disposed in an exhaust path of the internal combustion engine, the upstream catalyst storing a predetermined component of exhaust gas in an oxidizing atmosphere and releasing the predetermined component in a reducing atmosphere; a downstream catalyst disposed downstream of the upstream catalyst in the exhaust path, the downstream catalyst storing the predetermined component of the exhaust gas in an oxidizing atmosphere and releasing the predetermined component in a reducing atmosphere; and characterised by a controller adapted to execute a first purging control if a quantity of the predetermined component stored in the upstream catalyst becomes equal to or larger than a first threshold, the first purging control involving release of the predetermined component from the upstream catalyst through supply of a reductant, and to execute a second purging control if a quantity of the predetermined component stored in the downstream catalyst becomes equal to or larger than a second threshold, the second purging control involving release of the predetermined component from the downstream catalyst through supply of the reductant, wherein if the quantity of the predetermined component stored in the upstream catalyst exceeds a third threshold and if the quantity of the predetermined component stored in the downstream catalyst exceeds a fourth threshold, the controller simultaneously and continuously executes the first purging control and the second purging control, the third threshold being smaller than the first threshold, the fourth threshold being smaller than the second threshold.