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The controllability and observability properties of a singular system are extensively studied. The definitions of controllability,R-controllability, and impulse controllability are introduced via characteristics of the original state vector. Analogous definitions are presented for the case of observability. The criteria established for controllability and observability are simple rank criteria related to the Markov parameters from the inputs to the states and from the initial conditions to the outputs, respectively. The present results can be considered as the direct extension of Kalman's controllability and observability criteria to the case of singular systems. Finally, the controllability and observability subspaces are derived from the image and the kernel of the controllability and the observability matrices, respectively.