We introduce a method to construct G<inf>2</inf> –instantons over compact G<inf>2</inf> –manifolds arising as the twisted connected sum of a matching pair of building blocks. Our construction is based on gluing G<inf>2</inf> –instantons obtained from holomorphic vector bundles over the building blocks via the first author’s work. We require natural compatibility and transversality conditions which can be interpreted in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a K3 surface © 2015, Mathematical Sciences Publishers. All rights reserved.19312631285Atiyah, M., New invariants of 3– and 4–dimensional manifolds, from: “The mathematical heritage of Hermann Weyl”, (RO Wells, Jr, editor) (1988) Proc. Sympos. Pure Math. 48, Amer. Math. Soc, pp. 285-299. , MR974342Corti, A., Haskins, M., Nordström, J., Pacini, T., (2012) G<inf>2</inf> –manifolds and Associative Submanifolds via Semi-Fano 3–folds, , arXiv:1207.4470v2A Corti, M., Haskins, J., Nordström, T., Pacini, Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds (2013) Geom. Topol, 17, pp. 1955-2059. , MR3109862Demailly, J.-P., Complex Analytic and Differential Geometry, , https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdfDonaldson, S.K., Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles (1985) Proc. London Math. Soc, 50, pp. 1-26. , MR765366Donaldson, S.K., Floer homology groups in Yang–Mills theor (2002) Cambridge Tracts in Math, 147p. , Cambridge Univ. Press, MR1883043Donaldson, S.K., Kronheimer, P.B., (1990) The Geometry of Four-Manifolds, , Oxford Univ. Press, MR1079726Donaldson, S., Segal, E., Gauge theory in higher dimensions, II, from: “Geometry of special holonomy and related topics (2011) Surv. Differ. Geom, 16, pp. 1-41. , NC Leung, S-T Yau, editors), International Press, Boston, MR2893675Donaldson, S.K., Thomas, R.P., (1998) Gauge Theory in Higher Dimensions, From: “The Geometric universe, pp. 31-47. , (SA Huggett, L J Mason, KP Tod, S T Tsou, NMJ Woodhouse, editors), Oxford Univ. Press, MR1634503Earp, H., G<inf>2</inf> –instantons over asymptotically cylindrical manifolds (2015) Geom. Topol, 19, pp. 61-111Griffiths, P., Harris, J., (1994) Principles of Algebraic Geometry, , Wiley Classics Library, John Wiley & Sons, MR1288523Haskins, M., Hein, H.-J., Nordström, J., (2014) Asymptotically Cylindrical Calabi–Yau Manifolds, , arXiv:1212.6929v3Huybrechts, D., Lehn, M., The geometry of moduli spaces of sheaves (1997) Aspects of Math, , E31, Friedr. Vieweg & Sohn, MR1450870Joyce, D.D., Compact Riemannian 7–manifolds with holonomy G<inf>2</inf> : I, II (1996) J. Differential Geom, 43, pp. 291–328, 329–375. , MR1424428Joyce, D.D., (2000) Compact Manifolds with Special Holonomy, , Oxford Univ. Press, MR1787733Kovalev, A., Twisted connected sums and special Riemannian holonomy (2003) J. Reine Angew. Math, 565, pp. 125-160. , MR2024648Kovalev, A., N-H Lee, K3 surfaces with nonsymplectic involution and compact irreducible G<inf>2</inf> –manifolds (2011) Math. Proc. Cambridge Philos. Soc, 151, pp. 193-218. , MR2823130Lockhart, R.B., McOwen, R.C., Elliptic differential operators on noncompact manifolds (1985) Ann. Scuola Norm. Sup. Pisa Cl. Sci, 12, pp. 409-447. , MR837256Mazja, V.G., Plamenevskiĭ, B.A., Estimates in Lp and in Hölder classes, and the Miranda–Agmon maximum principle for the solutions of elliptic boundary value problems in domains with singular points on the boundary (1978) Math. Nachr, 81, pp. 25-82. , MR0492821Tyurin, A., (2008) Vector Bundles, , Univ. Göttingen, MR2742585Uhlenbeck, K., Yau, S.-T., On the existence of Hermitian–Yang–Mills connections in stable vector bundles (1986) Comm. Pure Appl. Math, 39, pp. S257-S293. , MR861491Walpuski, T., Gauge theory on G<inf>2</inf> –manifolds, PhD thesis (2013) Imperial College London, , http://tinyurl.com/o8zh9bnWalpuski, T., G<inf>2</inf> –instantons on generalised Kummer constructions, Geom (2013) Topol, 17, pp. 2345-2388. , MR3110581