A SURVEY OF DIOPHANTINE APPROXIMATION IN FIELDS OF POWER SERIES A. Lasjaunias The fields of power series (or perhaps better called formal numbers) are analogues of the field of real numbers. Many questions in number the- ory which have been studied in the setting of the real numbers can be transposed in the setting of the power series. The study of rational ap- proximation to algebraic real numbers has been intensively developped starting from the middle of the nineteenth century with the work of Liou- ville up to the celebrated theorem of Roth established in 1955. In the last thirty years, several mathematicians have studied diophantine approxima- tion in fields of power series. We present here a summary of the present knowledge on this subject, emphasizing the analogies and differences with the situation in the real numbers case. 1. The fields of power series . Let K be a field. If T is an indeterminate, we consider the ring K [T ] of polynomials in T with coefficients in K , and the field K (T ) of rational functions in T with coefficients in K . An ultrametric absolute value in K (T ) is defined by |0| = 0 and |P/Q| = |T | deg P −deg Q , where |T | is a fixed real number greater than 1. We consider the completion field of K (T ) for this absolute value, which is denoted K ((T −1 )) and called the field of power series over K . Then if α ∈ K ((T −1 )), and α = 0, we can write α =  k≤k 0 a k T k where k 0 ∈ Z , a k ∈ K and a k 0 =0. The degree of α = 0 is defined by extension as deg α = k 0 . So the absolute value is extended in K ((T −1 )), and we have |α| = |T | deg α if α = 0. This construction is clearly similar to the construction of the real numbers. The analogues of Z, Q and R are respectively K [T ], K (T ) and K ((T −1 )). Here we study the approximation of the elements of K ((T −1 )) by the elements of K (T ). Particularly we consider this approximation for the elements of K ((T −1 )) which are algebraic over K (T ). This analogy between the field of real numbers and the field of formal power series can be considered from another point of view. Indeed the sequence of the coefficients of a power series can be compared to the sequence of the digits in the decimal expansion of a real number. We illustrate here this parallelism with an analogue of a classical result for the real numbers. Let α ∈ K ((T −1 )). It is easy to prove that the sequence of the coefficients of the power series representing α is ultimately periodic if and only if there exist integers n ≥ 0 and m ≥ 1 such that T n (T m − 1) α ∈ 1