In the {\em distributed all-pairs shortest paths} problem (APSP), every node in the weighted undirected distributed network (the CONGEST model) needs to know the distance from every other node using least number of communication rounds (typically called {\em time complexity}). The problem admits $(1+o(1))$-approximation $\tildeΘ(n)$-time algorithm and a nearly-tight $\tilde Ω(n)$ lower bound [Nanongkai, STOC'14; Lenzen and Patt-Shamir PODC'15]\footnote{$\tilde Θ$, $\tilde O$ and $\tilde Ω$ hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios~\cite{LenzenP_podc13,HolzerW12,PelegRT12,Nanongkai-STOC14}.}. For the exact case, Elkin [STOC'17] presented an $O(n^{5/3} \log^{2/3} n)$ time bound, which was later improved to $\tilde O(n^{5/4})$ [Huang, Nanongkai, Saranurak FOCS'17]. It was shown that any super-linear lower bound (in $n$) requires a new technique [Censor-Hillel, Khoury, Paz, DISC'17], but otherwise it remained widely open whether there exists a $\tilde O(n)$-time algorithm for the exact case, which would match the best possible approximation algorithm. This paper resolves this question positively: we present a randomized (Las Vegas) $\tilde O(n)$-time algorithm, matching the lower bound up to polylogarithmic factors. Like the previous $\tilde O(n^{5/4})$ bound, our result works for directed graphs with zero (and even negative) edge weights. In addition to the improved running time, our algorithm works in a more general setting than that required by the previous $\tilde O(n^{5/4})$ bound; in our setting (i) the communication is only along edge directions (as opposed to bidirectional), and (ii) edge weights are arbitrary (as opposed to integers in {1, 2, ... poly(n)}). ...