Klein bottles and their kin were rarely spotted outside of the obscure corners of topology textbooks and fairy tales in popular math books -- until the dawn of Topological Data Analysis (TDA). In this talk, I will overview key ideas in TDA and share the results of my project. We will see fancy plots. We will encounter a Klein bottle seated in image datasets. We will discover tiny topological signals disguised as noise. This is a tale of courageous topologists leaving the comfort of idealized forms into the chaotic corporeal world to recover topological structures from corrupt data.