This memoir aims at introducing adaptation within the Multiple-Try Metropolis (MTM) algorithms which are a special case of the Markov chain Monte Carlo (MCMC) methods. The MCMC methods, along with their adaptive and multiple-try extensions, are thoroughly explored (both in their possible variations and in their theoretical properties) in order to firmly anchor the study of the proposed adaptive Multiple-Try Metropolis (aMTM) algorithm. Moreover, some existing results on the properties of MTM algorithms are generalized to enable more general results about the aMTM algorithm. The ergodicity of the algorithm is then established using well known results of Roberts and Rosenthal (2007), Andrieu and Moulines (2006) and Craiu et al. (2015) and its empirical performance is studied through a series of simulation experiments. The aMTM algorithm achieves notably better performance than simpler samplers (non-adaptive or single-try) when applied to distributions that are multimodal or that exhibit complex geometry. Finally, many variations of the algorithm are proposed and compared to identify settings that are particularly more efficient. An implementation of the algorithm is provided in a R package called aMTM available at https://github.com/fontaine618/aMTM.