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In view of potential applications of CubeSat technology to small-body planetary exploration, this paper studies the feasibility of using autonomous CubeSats to flyby near-Earth asteroids. This work provides an overview of the current state of CubeSat technology and proposes a 3U CubeSat mission using primarily off-the-shelf components. The proposed mission considers a CubeSat is deployed by a larger spacecraft in a periodic orbit around the first (L1) or the second (L2) Sun-Earth Lagrange points (common destinations to observe the Sun and outer space), from where fuel-optimal impulsive trajectories are designed to flyby asteroids between 2019 and 2025. Navigation support and ground operations costs still represent a major challenge for interplanetary CubeSats. As such, Monte Carlo simulations are performed to determine the flyby accuracies that can be accomplished by a 3U CubeSat flying autonomously (i.e., using observations of the Sun during cruise and observations of the asteroid before the flyby to estimate its own trajectory, instead of using ground stations for navigation support). Asteroid flyby opportunities for an autonomous 3U CubeSat are identified between years 2019 and 2025. Flyby altitudes below 500 km are found possible with currently-available CubeSat components. 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