Euclidean spanners are important geometric structures, having found numerous applications over the years. Cornerstone results in this area from the late 1980s and early 1990s state that for any $d$-dimensional $n$-point Euclidean space, there exists a $(1+\epsilon)$-spanner with $n \cdot O(\epsilon^{-d+1})$ edges and lightness (normalized weight) $O(\epsilon^{-2d})$. Surprisingly, the fundamental question of whether or not these dependencies on $\epsilon$ and $d$ for small $d$ can be improved has remained elusive, even for $d = 2$. This question naturally arises in any application of Euclidean spanners where precision is a necessity (thus $\epsilon$ is tiny). In the most extreme case $\epsilon$ is inverse polynomial in $n$, and then one could potentially improve the size and lightness bounds by factors that are polynomial in $n$. The state-of-the-art bounds $n \cdot O(\epsilon^{-d+1})$ and $O(\epsilon^{-2d})$ on the size and lightness of spanners are realized by the greedy spanner. In 2016, in a preliminary version, Filtser and Solomon [SIAM J. Comput., 49 (2020), pp. 429--447] proved that, in low-dimensional spaces, the greedy spanner is “near-optimal''; informally, their result states that the greedy spanner for dimension $d$ is just as sparse and light as any other spanner but for dimension larger by a constant factor. Hence the question of whether the greedy spanner is truly optimal remained open to date. The contribution of this paper is twofold: (1) We resolve these longstanding questions by nailing down the dependencies on $\epsilon$ and $d$ and showing that the greedy spanner is truly optimal. Specifically, for any $d= O(1), \epsilon = \Omega({n}^{-\frac{1}{d-1}})$, (a) we show that there are $n$-point sets in $\mathbb{R}^d$ for which any $(1+\epsilon)$-spanner must have $n \cdot \Omega(\epsilon^{-d+1})$ edges, implying that the greedy (and other) spanners achieve the optimal size; (b) we show that there are $n$-point sets in $\mathbb{R}^d$ for which any $(1+\epsilon)$-spanner must have lightness $\Omega(\epsilon^{-d})$, and then improve the upper bound on the lightness of the greedy spanner from $O(\epsilon^{-2d})$ to $O(\epsilon^{-d} \log(\epsilon^{-1}))$. (The lightness upper and lower bounds match up to a lower-order term.) (2) We then complement our negative result for the size of spanners with a rather counterintuitive positive result: Steiner points lead to a quadratic improvement in the size of spanners! Our bound for the size of Steiner spanners in $\mathbb{R}^2$ is tight as well (up to a lower-order term).