1. Introduction. Continued fractions [8,14] are a useful tool in many number theoretical problems and in numerical computing. It is well known that the simple continued fraction expansion of a single real number gives the best solution to its rational approximation problem. Many people have contrived to construct multidimensional continued fractions in dealing with the rational approximation problem for multi-reals. One construction is the Jacobi-Perron algorithm (JPA) (see [1]). This algorithm and its modifications have been extensively studied [6,7,10,13]. These algorithms are adapted to study the same problem for multi-formal Laurent series [2,4,11,12]. But none of them guarantees the best rational approximation in general. In this paper, we deal with the multi-rational approximation problem over the formal Laurent series field F ((z −1 )): given an element r ∈ F ((z −1 )) m , find p ∈ F [z] m and q ∈ F [z] such that p/q approximates r as close as possible while deg(q) is bounded.We propose a new continued fraction algorithm for multi-formal Laurent series. It is proved that this algorithm guarantees best rational approximations for multi-formal Laurent series.The paper is organized as follows: Section 2 deals with the indexed valuation of F ((z −1 )) m . Section 3 contains the detailed definition of the problem of optimal rational approximation of multi-formal Laurent series. Section 4 proposes an algorithm called multidimensional continued fraction algorithm (m-CFA, for short), which produces a multi-continued fraction expansion C(r) for any given multi-formal Laurent series r. Section 5 shows that C(r) satisfies three basic conditions. Section 6 states the main results of this pa-