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The proposal of increasingly complex and innovative space endeavours poses growing demands for mission designers. In order to meet the established requirements and constraints while maintaining a low fuel cost, the use of low-energy trajectories is particularly interesting. These allow spacecraft to change orbits and move with little to no fuel, but they are computed using motion models of a higher fidelity than the commonly used two-body problem (2BP). For this purpose, perturbation methods that explore the third-body effect are especially attractive, since they can accurately convey the system dynamics of a three-body configuration with a lower computational cost, by employing mapping techniques or exploring analytical approximations. The focus of this work is to broaden the knowledge of low-energy trajectories by developing new mathematical tools to assist in mission design applications. In particular, novel motion models based on the third-body effect are conceived. One application of this study focuses on the trajectory design for missions to near-Earth asteroids. Two different projects are explored: one is based on the preliminary design of separate rendezvous and capture missions to the invariant manifolds of libration point $L_2$. This is achieved by studying two recently discovered asteroids and determining dates, fuel cost and final control history for each trajectory. The other covers a larger study on asteroid capture missions, where several bodies are regarded as potential targets. The candidates are considered using a multi-fidelity design framework that filters through the trajectory options using models of motion of increasing accuracy, so that a final refined, low-thrust solution is obtained. The trajectory design hinges on harnessing Earth's gravity by exploiting encounters outside its sphere of influence, the named Earth-resonant encounters.