This talk explores a series of preservation principles concerning well-orders and well quasi-orders, wqo. Well-orders are foundational in mathematical logic, playing a central role in set theory—especially through the concept of ordinals—and in proof theory, notably in ordinal analysis, which aims to associate ordinals with formal theories. Given their importance, functions between well-orders are a natural subject of study, leading to the formulation of Well-Ordering Principles (WOP). For a function g from countable ordinals to countable ordinals, the principle WOP(g) formalizes the assertion: “For all α, if α is a well-order, then g(α) is a well-order.” Well quasi-orders are the quasi-order analog of well-orders and are fundamental in areas ranging from combinatorics to theoretical computer science. Correspondingly, suitable analogs of WOPs can be defined for wqos. Specifically, for a set operation G that preserves the wqo property (such as disjoint union), the Well Quasi-order closure Property WQP(G) formalizes the statement: “For all Q, if Q is a wqo, then G(Q) is a wqo.” In this talk, we present recent results concerning the ordinal and proof-theoretic analysis of WOPs and WQPs, focusing on variants of Kruskal’s Tree Theorem and operations such as the iterated union and product of wqos.