Abstract: Two types of concentric optical systems used in imaging spectrometers, Offner convex grating imaging spectrometers and Dyson concave grating imaging spectrometers, are introduced. The Offner convex grating imaging spectrometers are characterized by their broadly spectrum band, small influence from environments and easy to be fabricated and assembled, and the Dyson concave grating imaging spectrometers have advantages in their small volumes and easy to realize the compact systems. The design examples of the above types are given, and the results show that both the imaging spectrometers can offer high quality optical modulation transfer functions and can reach the perfect results in distortion less than 0.005% and spectrum resolution less than 3 nm. Moreover, several kind of foreoptics systems matched with the imaging spectrometers are given and the methods to eliminate the stray light and to restrain the spectrum overlaps in the optical systems are discussed.