The design of observers is investigated for the estimation of the descriptor vectorx(t)of descriptor systems of the formEx(t) = Ax(t) + Bu(t), y(t) = Cx(t).Such systems are divided into two categories: non-causal (impulsive) systems and causal (impulse- free) systems. For non-causal systems, an observer of order rank [E] is proposed and termed a full-order observer, since the dynamical order of a descriptor system is regarded as rank [E]. For causal systems, it is shown that the number of independent outputsmdoes not exceed rank [E], and an observer of order (rank [E] —m) is introduced and termed a minimal-order observer. The approach is based on the transformation of the system into particular design coordinates via a singular value decomposition of the matrixE.Illustrative examples are included.