We give a solution of the blow-up problem for equation u = e u , with data close to constants, in any number of space dimensions: there exists a blow-up surface, near which the solution has logarithmic behavior; its smoothness is estimated in terms of the smoothness of the data. More precisely, we prove that for any solution of u = e u with Cauchy data on t = 1 close to (ln 2, −2) in H s (R n) × H s−1 (R n), s is a large enough integer, must blow-up on a space like hypersurface defined by an equation t = ψ(x) with ψ ∈ H s−146−9[n/2] (R n). Furthermore, the solution has an asymptotic expansion ln(2/T 2) + j,k u jk (x)T j+k (ln T) k , where T = t − ψ(x), valid upto order s − 151 − 10[n/2]. Logarithmic terms are absent if and only if the blow-up surface has vanishing scalar curvature. The blow-up time can be identified with the infimum of the function ψ. Although attention is focused on one equation, the strategy is quite general; it consists in applying the Nash-Moser IFT to a map from " singularity data " to Cauchy data.