Identification of time-varying linear systems, which introduce both time-shifts (delays) and 
frequency-shifts (Doppler-shifts) to the input signal, is a central task in many engineering applications. 
This paper studies the problem of identification of underspread linear systems (ULSs), defined as 
time-varying linear systems whose responses lie within a unit-area region in the delay--Doppler space, by 
probing them with a known input signal and analyzing the resulting system output. One of the main 
contributions of the paper is a characterization of the conditions on the bandwidth and temporal support of 
the input signal that ensure identification of a ULS described by a finite set of delays and Doppler-shifts, 
and referred to as a parametric ULS, from a single observation. In particular, the paper establishes that 
sufficiently-underspread parametric linear systems are identifiable as long as the time--bandwidth product 
of the input signal is proportional to the square of the total number of delay--Doppler pairs in the system. 
In addition, an algorithm is developed that enables identification of parametric ULSs from an input train of 
pulses in polynomial time by exploiting recent results on sub-Nyquist sampling for time-delay estimation and 
classical results on recovery of frequencies from a sum of complex exponentials. Finally, application of 
these results to super-resolution target detection using radar is discussed. Specifically, it is shown that 
the proposed procedure allows to distinguish between multiple targets with very close proximity in the 
delay--Doppler space, resulting in a resolution that substantially exceeds that of standard 
matched-filtering-based techniques without introducing leakage effects inherent in recently proposed 
compressed sensing-based radar methods.