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DES Collaboration: L. F. Secco et al., We present high signal-to-noise measurements of three-point shear correlations and the third moment of the mass aperture statistic using the first 3 years of data from the Dark Energy Survey. We additionally obtain the first measurements of the configuration and scale dependence of the four three-point shear correlations which carry cosmological information. With the third-order mass aperture statistic, we present tomographic measurements over angular scales of 4 to 60 arcminutes with a combined statistical significance of 15.0σ. Using the tomographic information and measuring also the second-order mass aperture, we additionally obtain a skewness parameter and its redshift evolution. We find that the amplitudes and scale-dependence of these shear 3pt functions are in qualitative agreement with measurements in a mock galaxy catalog based on N-body simulations, indicating promise for including them in future cosmological analyses. We validate our measurements by showing that B-modes, parity-violating contributions and PSF modeling uncertainties are negligible, and determine that the measured signals are likely to be of astrophysical and gravitational origin., M. J. is supported in part by National Science Foundation Grant No. 1907610. B. J. is supported in part by the U.S. Department of Energy Grant No. DE-SC0007901. C. C. is supported by DOE grant DE-SC0021949. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministerio da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energeticas, Medioambientales y Tecnológicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciencies de l’Espai (IEEC/CSIC), the Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, Texas A&M University, and the OzDES Membership Consortium. The DES data management system is supported by the National Science Foundation under Grants No. AST-1138766 and No. AST-1536171. The DES participants from Spanish institutions are partially supported by MICINN under Grants No. ESP2017-89838, No. PGC2018-094773, No. PGC2018-102021, No. SEV-2016-0588, No. SEV-2016-0597, and No. MDM-2015-0509, some of which include ERDF funds from the European Union. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya. Research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Program (FP7/2007-2013) including ERC Grants agreements No. 240672, No. 291329, and No. 306478. We acknowledge support from the Brazilian Instituto Nacional de Ciência e Tecnologia (INCT) do e-Universo (CNPq Grant No. 465376/2014-2). We acknowledge support from the Australian Research Council Centre of Excellence for Allsky Astrophysics (CAASTRO), through Project No. CE110001020. This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DEAC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics.