This paper considers the problem of identifiability and parameter estimation of single-input-single-output, linear, time-invariant, stable, continuous-time systems under irregular and random sampling schemes. Conditions for system identifiability are established under inputs of exponential polynomial types and a tight bound on sampling density. Identification algorithms of Gauss-Newton iterative types are developed to generate convergent estimates. When the sampled output is corrupted by observation noises, input design, sampling times, and convergent algorithms are intertwined. Persistent excitation (PE) conditions for strongly convergent algorithms are derived. Unlike the traditional identification, the PE conditions under irregular and random sampling involve both sampling times and input values. Under the given PE conditions, iterative and recursive algorithms are developed to estimate the original continuous-time system parameters. The corresponding convergence results are obtained. Several simulation examples are provided to verify the theoretical results. (C) 2015 Elsevier Ltd. All rights reserved.