We present improved deterministic distributed algorithms for a number of well-studied matching problems, which are simpler, faster, more accurate, and/or more general than their known counterparts. The common denominator of these results is a deterministic distributed rounding method for certain linear programs, which is the first such rounding method, to our knowledge. A sampling of our end results is as follows. -- An $O(\log^2 Δ\cdot \log n)$-round deterministic distributed algorithm for computing a maximal matching, in $n$-node graphs with maximum degree $Δ$. This is the first improvement in about 20 years over the celebrated $O(\log^4 n)$-round algorithm of Hańćkowiak, Karoński, and Panconesi [SODA'98, PODC'99]. -- A deterministic distributed algorithm for computing a $(2+\varepsilon)$-approximation of maximum matching in $O(\log^2 Δ\cdot \log \frac{1}{\varepsilon} + \log^ * n)$ rounds. This is exponentially faster than the classic $O(Δ+\log^* n)$-round $2$-approximation of Panconesi and Rizzi [DIST'01]. With some modifications, the algorithm can also find an $\varepsilon$-maximal matching which leaves only an $\varepsilon$-fraction of the edges on unmatched nodes. -- An $O(\log^2 Δ\cdot \log \frac{1}{\varepsilon} + \log^ * n)$-round deterministic distributed algorithm for computing a $(2+\varepsilon)$-approximation of a maximum weighted matching, and also for the more general problem of maximum weighted $b$-matching. These improve over the $O(\log^4 n \cdot \log_{1+\varepsilon} W)$-round $(6+\varepsilon)$-approximation algorithm of Panconesi and Sozio [DIST'10], where $W$ denotes the maximum normalized weight.