A cetane number estimating apparatus characterized by comprising: a plurality of cylinders (#1-#4) provided for an internal combustion engine which can re-circulate one portion of exhaust gas from an exhaust system to an intake system; an injecting device for injecting fuel into the cylinders; a measuring device for measuring ignition timing in one cylinder on the basis of a change in pressure in the one cylinder; a judging device for judging whether or not the fuel is burnt in another cylinder which is in an exhaust process when one cylinder is in an intake process; and an estimating device for estimating the cetane number of the injected fuel on the basis of the measured ignition timing and a first correlation between the cetane number and the ignition timing corresponding to a case where non-combustion gas in which the fuel is not burnt is emitted by said another cylinder, when the one cylinder is in the intake process, if it is judged that the fuel is not burnt in said another cylinder; and for estimating the cetane number of the injected fuel on the basis of the measured ignition timing and a second correlation between the cetane number and the ignition timing corresponding to a case where combustion gas in which the fuel is burnt is emitted by said another cylinder, when the one cylinder is in the intake process, if it is judged that the fuel is burnt in said another cylinder, wherein the estimating device estimates the cetane number on the basis of the first correlation, if fuel cut is performed in a first period which is a time interval from timing of a first compression process in the one cylinder to timing of a second compression process in said another cylinder which performs the second compression process next to the one cylinder, the estimating device estimates the cetane number on the basis of the second correlation, if fuel cut is performed in a second period which is a time interval from timing of a second compression process in said another cylinder which performs the second compression process next to a first compression process in the one cylinder; to timing of a third compression process performed next to the first compression process in the one cylinder. a plurality of cylinders (#1-#4) provided for an internal combustion engine which can re-circulate one portion of exhaust gas from an exhaust system to an intake system; an injecting device for injecting fuel into the cylinders; a measuring device for measuring ignition timing in one cylinder on the basis of a change in pressure in the one cylinder; a judging device for judging whether or not the fuel is burnt in another cylinder which is in an exhaust process when one cylinder is in an intake process; and an estimating device for estimating the cetane number of the injected fuel on the basis of the measured ignition timing and a first correlation between the cetane number and the ignition timing corresponding to a case where non-combustion gas in which the fuel is not burnt is emitted by said another cylinder, when the one cylinder is in the intake process, if it is judged that the fuel is not burnt in said another cylinder; and for estimating the cetane number of the injected fuel on the basis of the measured ignition timing and a second correlation between the cetane number and the ignition timing corresponding to a case where combustion gas in which the fuel is burnt is emitted by said another cylinder, when the one cylinder is in the intake process, if it is judged that the fuel is burnt in said another cylinder, wherein the estimating device estimates the cetane number on the basis of the first correlation, if fuel cut is performed in a first period which is a time interval from timing of a first compression process in the one cylinder to timing of a second compression process in said another cylinder which performs the second compression process next to the one cylinder, the estimating device estimates the cetane number on the basis of the second correlation, if fuel cut is performed in a second period which is a time interval from timing of a second compression process in said another cylinder which performs the second compression process next to a first compression process in the one cylinder; to timing of a third compression process performed next to the first compression process in the one cylinder.