  We examine several algebraic properties of the noncommutive $z$-plane and
Riemann surfaces. The starting point of our investigation is a two-dimensional
noncommutative field theory, and the framework of the theory will be converted
into that of a complex coordinate system. The basis of noncommutative complex
analysis is obtained thoroughly, and the considerations on functional analysis
are also given before performing the examination of the conformal mapping and
the Teichm\"{u}ller theory. (Keywords; Complex Analysis, Riemann Surfaces and
Teichm\"{u}ller Space, Functional Analysis, Deformation Quantization,
Non-Commutative Geometry, Quantum Groups)
