Moduli spaces and moduli stacks of vector bundles and of principal $G$-bundles for a complex algebraic group $G$ have been widely studied in geometry, with many deep results. In a classical work of Grothendieck presented in FGA (see FGA explained), moduli schemes of coherent sheaves with certain parameters fixed, so called Quot schemes. Later the geometric invariant theory defined other class of moduli spaces of bundles. Especially important is the case of moduli space of stable bundles on a Riemann surface which is extremely important for mathematical physics (self-dual solutions of Yang-Mills equations; study of spaces of conformal blocks; representation theory of affine Lie algebras and loop groups; integrable systems, esp. Hitchin system etc.).
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G. Faltings, Vector bundles on curves, 1995 Bonn lectures, write up by M. Stoll, pdf
G. Faltings, Moduli-stacks for bundles on semistable curves, Math. Ann. 304, 3 (1996) 489-515; Stable $G$-bundles and projective connections, J. Algebraic Geom. 2, 3 (1993) 507-568, doi, Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. 5 (2003), 41-68.
G. Faltings, Line-bundles on the moduli-space of G-torsors, lecture at MSRI 2002, video and pdf
G. Harder, M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundle on curves, Math Ann. 212, 215-248 (1975).
N.J. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987) pp. 91–114
M. S. Narasimhan, C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. Math. 82, No. 3 (Nov., 1965), pp. 540-567, jstor, doi
V. B. Mehta, C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), 205–239.
P. E. Newstead, Characteristic classes of stable bundles of rank 2 over an algebraic curve, Trans. Amer. Math. Soc. 169 (1972), 337–345.
A. Ramanathan, Stable principal bundles on a compact Riemann surface, Math Ann 213, 129-152 (1975).
S. Ramanan, The moduli spaces of vector bundles over an algebraic curve, Math. Ann. 200 (1973), 69–84.
Carlos T. Simpson, Higgs bundles and local systems, Publ. Mathématiques de l’IHÉS 75 (1992), p. 5-95, numdam
C. Teleman, C. T. Woodward, The index formula for the moduli of G-bundles on a curve, Ann. Math. 170, 2, 495–527 (2009) pdf
References for moduli spaces of bundles over singular curves are discussed at MathOverflow here