  The global conformal gauge is playing the crucial role in string theory
providing the basis for quantization. Its existence for two-dimensional
Lorentzian metric is known locally for a long time. We prove that if a
Lorentzian metric is given on a plain then the conformal gauge exists globally
on the whole ${\mathbb R}^2$. Moreover, we prove the existence of the conformal
gauge globally on the whole worldsheets represented by infinite strips with
straight boundaries for open and closed bosonic strings. The global existence
of the conformal gauge on the whole plane is also proved for the positive
definite Riemannian metric.
