The purpose of this opening lecture is to ground the participants in a few of the basic definitions of cosheaf theory and introduce some open problems. By studying sheaves and cosheaves associated to stratified maps, I'll propose a definition in terms of assignments to basic open sets and state a result in terms of Kan extensions. This perspective will cover many of the examples of interest to the workshop. By considering the down-set topology on the real line, I'll consider a different perspective on merge trees and state some open problems for the working groups related to Reeb graphs. One of the open problems of interest will be to devise an "Elder rule" for the level-set barcode associated to a Reeb graph.