Algebraic $K$-theory is an invariant that assigns to a ring $R$ groups $K_n(R)$ for each nonnegative integer $n$. These $K$-groups are interesting for their connections to number theory, algebraic geometry, and geometric and algebraic topology, but they are also notoriously difficult to compute. This ubiquity is also their undoing: $K$-theory is in some sense a universal invariant.

In this talk, I will explain a construction of algebraic $K$-theory and explain some reasons why computations are difficult. I will also describe techniques that are used to nevertheless extract information about $K$-theory.