When the results of certain computer calculations are shown to be not simply incorrect but dramatically incorrect, we have a powerful reason to be cautious about all computer-based calculations. In this paper we present a "Rogue's Gallery" of simple calculations whose correct solutions are obvious to humans but whose numerical solutions are incorrect in pathological ways. We call these calculations, which can be guaranteed to wreak numerical mayhem across both software packages and hardware platforms, "Numerical Monsters". Our monsters can be used to provide deep insights into how computer calculations fail, and we use them to engender appreciation for the subject of numerical analysis in our students. Although these monsters are based on well-understood numerical pathologies, even experienced numerical analysts will surprises in their behaviour and can use the lessons they bring to become even better masters of their tools.