Ji, YuxiangThis thesis is divided into two parts. The first part focuses on theoretical problems arising from the study of K\"{a}hler edge geometry. The second part introduces a practical method for finding K\"{a}hler--Einstein and soliton metrics that exist on toric Fano surfaces. This method is based on a numerical implementation of the Ricci iteration, which essentially solves a sequence of Monge--Amp\`{e}re equations. Let $M$ be a compact \K manifold and $D=D_1+\dots D_r$ be a simple normal crossing divisor. Given that $K_M+D$ is ample, there exist \KE crossing edge metrics on $M$ with edge singularities of cone angle $\beta_i$ along each component $D_i$ for small $\beta_i$. The first result in the thesis shows that such negatively curved K\"{a}hler--Einstein crossing edge metrics converge to the K\"{a}hler--Einstein mixed cusp and edge metrics smoothly away from the divisor, as some of the cone angles approach $0$. We further show that, near the divisor, a family of appropriately renormalized K\"{a}hler--Einstein crossing edge metrics converges to a mixed cylinder and edge metric in the pointed Gromov--Hausdorff sense as some of the cone angles approach $0$ at (possibly) different speeds. Generalizing $\mathbb{P}^1$, Calabi--Hirzebruch manifolds are constructed by adding an infinite section to the total space of a tensor product of the hyperplane bundle over the projective space, leading to two disjoint divisors: the zero and infinite sections. The second main result in the thesis is the discovery of K\"{a}hler--Einstein edge metrics with singularities along the two divisors on Calabi--Hirzebruch manifolds, and the study on Gromov--Hausdorff limits of these metrics when either cone angle tends to its extreme value. As a very special case, we show that the Eguchi--Hanson metric arises in this way naturally as a Gromov--Hausdorff limit. We also completely describe all other (possibly rescaled) Gromov--Hausdorff limits which exhibit a wide range of behaviors, resolving in this setting a conjecture of Cheltsov--Rubinstein. This gives a new interpretation of both the Eguchi--Hansonspace and Calabi’s Ricci flat spaces as limits of compact singular Einstein spaces. The second part of the thesis focuses on numerical implementation of the Ricci iteration on toric del Pezzo surfaces: $\mathbb{P}^2$, $\mathbb{P}^1\times \mathbb{P}^1$, and blow-up of $\mathbb{P}^2$ at one, two or three distinct points in general position. The Ricci iteration on these surfaces can be reduced to solving a sequence of real Monge--Amp\`{e}re equations in $\mathbb{R}^2$ with the second boundary value condition. As the third contribution of the thesis, we successfully conduct the Ricci iteration on the aforementioned surfaces. We find that the resulting solutions numerically converge to either the unique K\"{a}hler--Einstein metric on $\mathbb{P}^2, \mathbb{P}^1\times \mathbb{P}^1$, and $\dPThr$, or the unique K\"{a}hler--Ricci soliton metric on $\dPOne$ and $\dPTwo$. This provides a novel numerical approach to finding K\"{a}hler--Einstein and soliton metrics on these manifolds. Our numerical results also provide evidence that, in the toric case, the Ricci iteration may converge and produce canonical metrics without the necessity of modifying the metrics obtained during the iterations by automorphisms.enDissertationMathematics