The concept of infinite-time admissibility of unbounded observation functionals is introduced. Under the assumption of exponential stability of the semigroup, it is equivalent to finite-time admissibility recently investigated in Weiss (1988 b, 1989). Necessary and sufficient criteria for admissibility are given. In particular, it is shown that the Ho-Russell-Weiss (1988 b) test for admissibility of observation functionals/control vectors can be derived in an elementary way without invoking the geometric interpretation of the Carleson measure, while the criterion in Weiss (1991) can easily be deduced from the Carleson Embedding Theorem. Some practically applicable sufficient conditions guaranteeing admissibility are discussed in 3. The results are illustrated by a feedback system containing RLCG transmission line.