Let $G = (V,E,w)$ be a weighted undirected graph on $|V| = n$ vertices and $|E| = m$ edges, let $k \ge 1$ be any integer, and let $ε< 1$ be any parameter. We present the following results on fast constructions of spanners with near-optimal sparsity and lightness, which culminate a long line of work in this area. (By near-optimal we mean optimal under Erdős' girth conjecture and disregarding the $ε$-dependencies.) - There are (deterministic) algorithms for constructing $(2k-1)(1+ε)$-spanners for $G$ with a near-optimal sparsity of $O(n^{1/k} \log(1/ε)/ε))$. The first algorithm can be implemented in the pointer-machine model within time $O(mα(m,n) \log(1/ε)/ε) + SORT(m))$, where $α( , )$ is the two-parameter inverse-Ackermann function and $SORT(m)$ is the time needed to sort $m$ integers. The second algorithm can be implemented in the WORD RAM model within time $O(m \log(1/ε)/ε))$. - There is a (deterministic) algorithm for constructing a $(2k-1)(1+ε)$-spanner for $G$ that achieves a near-optimal bound of $O(n^{1/k}\mathrm{poly}(1/ε))$ on both sparsity and lightness. This algorithm can be implemented in the pointer-machine model within time $O(mα(m,n) \mathrm{poly}(1/ε) + SORT(m))$ and in the WORD RAM model within time $O(m α(m,n) \mathrm{poly}(1/ε))$. The previous fastest constructions of $(2k-1)(1+ε)$-spanners with near-optimal sparsity incur a runtime of is $O(\min\{m(n^{1+1/k}) + n\log n,k n^{2+1/k}\})$, even regardless of the lightness. Importantly, the greedy spanner for stretch $2k-1$ has sparsity $O(n^{1/k})$ -- with no $ε$-dependence whatsoever, but its runtime is $O(m(n^{1+1/k} + n\log n))$. Moreover, the state-of-the-art lightness bound of any $(2k-1)$-spanner is poor, even regardless of the sparsity and runtime.