This limit is sometimes appearing in its own right, but sometimes it is just considered as an approximation for a system with fixed finite $N$. One of the features is that in the large N limit is that non-planar Feynman diagrams lose their importance and that the correlation functions satisfy certain decoupling/factorization rule. The behaviour is studied in terms of expansion in $1/N$ whose square has a simialr role to Planck constant in semiclassical approximation limit of quantum mechanics.
S. Coleman, 1/N, in Aspects of Symmetry, Cambridge University Press 1985.
A. V. Manohar, Large N QCD, L:es Houches Lecture 2004, pdf
A. Jevicki, Instantons and the $1/N$ expansion in nonlinear $\sigma$ models, Phys. Rev. D 20, 3331–3335 (1979) pdf
Juan M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv.Theor.Math.Phys.2:231-252, 1998 hep-th/9711200; Wilson loops in large N field theories, hep-th/9803002.
E. Brézin, S.R. Wadia, eds. The Large N Expansion in Quantum Field Theory and Statistical Physics, a book collection of reprinted historical articles, gBooks
Yuri Makeenko, Methods of contemporary gauge theory, Cambridge Monographs on Math. Physics, gBooks
M. Bershadsky, Z. Kakushadze, C. Vafa, String expansion as large N expansion of gauge theories, Nucl.Phys. B523 (1998) 59-72hep-th/9803076, doi
G.T. Horowitz, H. Ooguri, Spectrum of large N gauge theory from supergravity, hep-th/9802116
Semyon Klevtsov, Random normal matrices, Bergman kernel and projective embeddings, arxiv/1309.7333