This will be a gentle introduction to Geometric Group Theory. We will define the coarse geometry of a finitely generated group, and talk about growth rate as a coarse geometric invariant. We will give a brief sketch of Gromov's Polynomial Growth Theorem. This is a prototype "geometric hypotheses with algebraic conclusions" theorem. Asymptotic cones appear in the proof. Then we talk about the Word Problem for finitely presented groups, and introduce the Dehn function as another coarse geometric invariant related to the complexity of the Word Problem. Snowflake groups first appear as examples showing that the Dehn spectrum is dense in the interval \([2,\,\infty)\).

Finally, we talk about joint work with Hoda and Woodhouse showing that there are snowflake groups \(G\) such that every asymptotic cone of \(G\) is simply connected, but some asymptotic cone of \(G\) contains an isometrically embedded circle. These are the first examples of groups with this property.