Conditions are given for the existence of observers that estimate unmeasured outputs on the basis of partial information on the input and the state. The concepts of strong detectability and strong observability, introduced before in the literature for discrete-time systems only, are defined and studied for continuous-time systems. It is shown that there are two different concepts of strong detectability, which coincide for discrete-time systems. Algebraic conditions for either concept are given. It is shown that these concepts are intimately related to the existence of strong observers, i.e. observers that only use the output of the plant.