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The outer Galaxy is an environment with metallicity lower than the Solar one. Because of this, the formation and survival of molecules in star-forming regions located in the inner and outer Galaxy is expected to be different. To gain understanding on how chemistry changes throughout the Milky Way, it is crucial to observe outer Galaxy star-forming regions to constrain models adapted for lower metallicity environments. In this paper we present a new observational project: chemical complexity in star-forming regions of the outer Galaxy (CHEMOUT). The goal is to unveil the chemical composition in 35 dense molecular clouds associated with star-forming regions of the outer Galaxy through observations obtained with the IRAM 30m telescope. In this first paper, we present the sample, and report the detection at 3~mm of simple organic species HCO+, H13CO+, HCN, c-C3H2, HCO, C4H, and HCS+, of the complex hydrocarbon CH3CCH, and of SiO, CCS and SO. From c-C3H2, we estimate new kinematic heliocentric and Galactocentric distances based on an updated rotation curve of the Galaxy. The detection of the molecular tracers does not seem to have a clear dependence on the Galactocentric distance. We also analyse the HCO+ line profiles. We find high velocity wings in ~71% of the targets, and their occurrence does not depend on the Galactocentric distance. Our results, confirmed by a statistical analysis, show that the presence of organic molecules and tracers of protostellar activity is ubiquitous in the low-metallicity environment of the outer Galaxy. Based on this, and on the additional evidence that small, terrestrial planets are omnipresent in the Galaxy, we support previous claims that the definition of Galactic Habitable Zone should be rediscussed in view of the ubiquitous capacity of the interstellar medium to form organic molecules.