There has been significant recent work on the limits of covert communications, where the presence of a transmitted signal is kept hidden from an attentive adversary. In this paper, we turn our attention to the covert sensing problem. In particular, one of the main drawbacks of active sensing is that it can reveal the presence and/or location of the sensor. But active sensing is necessary or desirable in many applications, spanning areas from range finding in radar systems to channel estimation in wireless communication systems. Hence, we consider the active sensing of linear systems operating on the transmitted signal, but with the requirement that the probing signal is not detectable by a capable and attentive adversary. We first consider the frequency-nonselective standard 'block fading' channel from wireless communications and identify the conditions under which covert active sensing is possible. We then turn our attention to the covert sensing of linear systems modeled as bandlimited wide-sense stationary random processes, where we present initial but not conclusive results. Finally, we give thoughts on the extension of the results to frequency-selective models and present ideas for future work.