A soft-sided insulated container assembly is made of a flexible wall structure that is movable between a collapsed position and a deployed position. In one variation of the deployed position it resembles a tote-bag; in another variation of folding the deployed position corresponds to a generally box-like shape. The container includes a self-inflating wall structure that has an inner layer or skin, an outer layer or skin, and a resilient, open-cell insulating layer trapped between the two skins. A valve governs the ability to take in or to expel air. The outer layer may be thicker and more robust that the inner layer, the better to resist abrasion or punctures. The inner layer may be reflective. There may be a removable internal liner. That liner may be transparent.