  We describe the basic complex geometry and function theory of the {\em
pentablock} $\mathcal{P}$, which is the bounded domain in $\mathbb{C}^3$ given
by \[ \mathcal{P}= \{(a_{21}, \mathrm{tr} A, \det A): A= \begin{bmatrix}
a_{ij}\end{bmatrix}_{i,j=1}^2 \in \mathbb{B}\} \] where $\mathbb{B}$ denotes
the open unit ball in the space of $2\times 2$ complex matrices. We prove
several characterizations of the domain. We describe its distinguished boundary
and exhibit a $4$-parameter group of automorphisms of $\mathcal{P}$. We show
that $\mathcal{P}$ is intimately connected with the problem of $\mu$-synthesis
for a certain cost function $\mu$ on the space of $2\times 2$ matrices defined
in connection with robust stabilization by control engineers. We demonstrate
connections between the function theories of $\mathcal{P}$ and $\mathbb{B}$. We
show that $\mathcal{P}$ is polynomially convex and starlike.
