Despite significant research efforts, the state-of-the-art algorithm for maintaining an approximate matching in fully dynamic graphs has a polynomial {worst-case} update time, even for very poor approximation guarantees. In a recent breakthrough, Bhattacharya, Henzinger and Nanongkai showed how to maintain a constant approximation to the minimum vertex cover, and thus also a constant-factor estimate of the maximum matching size, with polylogarithmic worst-case update time. Later (in SODA'17 Proc.) they improved the approximation factor all the way to $2+ε$. Nevertheless, the longstanding fundamental problem of {maintaining} an approximate matching with sub-polynomial worst-case time bounds remained open. We present a randomized algorithm for maintaining an {almost-maximal} matching in fully dynamic graphs with polylogarithmic worst-case update time. Such a matching provides $(2+ε)$-approximations for both the maximum matching and the minimum vertex cover, for any $ε> 0$. Our result was done independently of the $(2+ε)$-approximation result of Bhattacharya et al., so it provides the first $(2+ε)$-approximation for minimum vertex cover (together with Bhattacharya et al.'s result) and the first $(2+ε)$-approximation for maximum (integral) matching. The polylogarithmic worst-case update time of our algorithm holds deterministically, while the almost-maximality guarantee holds with high probability. This result not only settles the aforementioned problem on dynamic matchings, but also provides essentially the best possible approximation guarantee for dynamic vertex cover (assuming the unique games conjecture).