The asymptotic theory of holomorphic linear differential equations of one variable is well understood and has recently been renewed by the theory of multisummation. However, the asymptotic theory of holonomic differential systems of many complex variables is still not completely developed. This volume tries to fill the gap by introducing the fundamental notions and by showing some consequences of such a theory. The notion of a good formal structure for a meromorphic vector bundle with a flat connection on a complex analytic surface is introduced. The existence of such a good formal structure on the pull-back by a suitable sequence of complex blowing-up of a meromorphic connection is conjectured. Some consequences of this conjecture are given : semi-continuity of the Malgrange-Komatsu irregularity index for an integrable family of meromorphic connections on a complex curve, and the construction and some properties of the Stokes fibration. The proof of the conjecture is given, among others, for bundles of rank $\leq 5$. The existence of a lifting of a good formal structure at the level of asymptotic expansions in bisectors is also shown. Applications are given to complex conjugation of holonomic $\mathcal {D}$-modules.