The classification of objects in algebra and geometry is of great interest to mathematicians. However, as early as the 1940's, results began to emerge showing that such things were sometimes impossible to do in an "algorithmic" sense. In this talk I will explain the notion of an algorithmic process, through the use of Turing Machines, and follow on by describing some questions in algebra and geometry that are in general undecidable (that is, admit no algorithmic solution). No background in decision problems will be assumed, and only a minimal amount of group theory and geometry will be needed to understand the examples (group presentations, manifolds).