We present the first data structures that maintain near optimal maximum cardinality and maximum weighted matchings on sparse graphs in sublinear time per update. Our main result is a data structure that maintains a $(1+ε)$ approximation of maximum matching under edge insertions/deletions in worst case $O(\sqrt{m}ε^{-2})$ time per update. This improves the 3/2 approximation given in [Neiman,Solomon,STOC 2013] which runs in similar time. The result is based on two ideas. The first is to re-run a static algorithm after a chosen number of updates to ensure approximation guarantees. The second is to judiciously trim the graph to a smaller equivalent one whenever possible. We also study extensions of our approach to the weighted setting, and combine it with known frameworks to obtain arbitrary approximation ratios. For a constant $ε$ and for graphs with edge weights between 1 and N, we design an algorithm that maintains an $(1+ε)$-approximate maximum weighted matching in $O(\sqrt{m} \log N)$ time per update. The only previous result for maintaining weighted matchings on dynamic graphs has an approximation ratio of 4.9108, and was shown in [Anand,Baswana,Gupta,Sen, FSTTCS 2012, arXiv 2012].