minors and topological minors are two closely related graph containment relations that have attracted extensive attentions. though giant breakthroughs on graph minors have been made over decades, several questions about these two relations remain open, especially for topological minors. this talk addresses part of our recent work in this direction, including a proof of robertson's conjecture on well-quasi-ordering graphs by the topological minor relation, a complete characterization of the graphs having the erdos-posa property with respect to topological minors which answers a question of robertson and seymour, and a proof of thomas' conjecture on half-integral packing. more open questions, such as hadwiger's conjecture on graph coloring and its variations and relaxations, will be discussed in this talk.