We propose a set of variables of the general three-body problem both for two-dimensional and three-dimensional cases. The variables are ($\lambda, \theta, \Lambda, \Theta, k, \omega$), or equivalently ($\lambda, \theta, L, \dot{I},k,\omega)$ for the two-dimensional problem, and ($\lambda, \theta, L, \dot{I}, k, \omega, \phi, \psi$) for the three-dimensional problem. Here, ($\lambda, \theta$) and ($\Lambda, \Theta)$ specify the positions in the shape spheres in the configuration and momentum spaces, $k$ is the virial ratio, $L$ is the total angular momentum, $\dot{I}$ is the time derivative of the moment of inertia, and $\omega, \phi$, and $\psi$ are the Euler angles to bring the momentum triangle from the nominal position to a given position. This set of variables defines a shape space of the three-body problem. This is also used as an initial-condition space. The initial condition of the so-called free-fall three-body problem is ($\lambda, \theta, k =$ 0, $L =$ 0, $\dot{I} =$ 0, $\omega =$ 0). We show that the hyper-surface $\dot{I} =$ 0 is a global surface of section.