How and why to embed surfaces in 4-dimensional spaces

Geometric topologists like to study spaces of arbitrary dimensions. Fortunately, we at least limit ourselves to studying manifolds, which locally mimic Euclidean space. Dimension four forms a "phase transition" between low- and high-dimensional manifolds, exhibiting unique behaviour and necessitating bespoke tools. I will describe the source of this curious phenomenon, giving a few guiding examples and constructions. The key source of the problem or appeal, depending on your perspective, of 4-dimensional manifolds turns out to be the difficulty in embedding surfaces therein.