Explore extremal singularities in algebraic geometry through this colloquium lecture delivered by Karen E. Smith at the 2022 Joint Mathematics Meetings. Delve into hypersurface singularities, examining their properties and the concept of the smallest possible singularity index. Investigate cone singularities and homogeneous polynomials, with a focus on cubic cones of dimension two. Learn about the classification and geometry of extremal singularities, which occur only in characteristic p > 0. Gain insights into the geometry of smooth cubic surfaces, including triangles on cubic surfaces and conditions for extremal singularities in cones over cubic surfaces. Discover how similar phenomena extend to higher degrees in this comprehensive exploration of advanced mathematical concepts.