The authors introduce a new set of global parameters, which they call pleating coordinates, for the Teichmüller space $T_{1,1}$  of the punctured torus. The space $T_{1,1}$ is embedded as a holomorphic family $G_\mu$ of Kleinian groups, where the complex parameter μ varies in a simply connected domain M in the complex plane. This is done in such a way that the regular set $\Omega (G_\mu )$ has a unique invariant component $\Omega _0(G_\mu )$ and the points in $T_{1,1}$ are represented by the Riemann surface $\Omega (G_\mu)/G_\mu$. This embedding is in fact the Maskit embedding. The new coordinates are geometric in the sense that they are related to the geometry of the hyperbolic manifold $H^3/G_\mu$. More precisely, they can be read off from the geometry of the punctured torus $\partial C_0/G_\mu$, where $\partial C_0$ is the component of the convex hull boundary facing $\Omega _0(G_\mu)$. The surface $\partial C_0$ has a natural hyperbolic metric and is pleated along geodesics that project to a geodesic lamination $\lambda$ on $\partial C_0/G_\mu$. There is some specific choice of transverse measure for the pleating lamination $\lambda$, which allows the authors to introduce a notion of pleating length for $G_\mu$. The laminations and their pleating lengths are the coordinates for $M$.