  The main purpose of the present paper is to introduce the notion of squeezing
functions of bounded domains and study some properties of them. The relation to
geometric and analytic structures of bounded domains will be investigated.
Existence of related extremal maps and continuity of squeezing functions are
proved. Holomorphic homogeneous regular domains are exactly domains whose
squeezing functions have positive lower bounds. Completeness of certain
intrinsic metrics and pseudoconvexity of holomorphic homogeneous regular
domains are proved by alternative method. In dimension one case, we get a neat
description of boundary behavior of squeezing functions of finitely connected
planar domains. This leads to a necessary and sufficient conditions for a
finitely connected planar domain to be a holomorphic homogeneous regular
domain. Consequently, we can recover some important results in complex
analysis. For annuli, we obtain some interesting properties of their squeezing
functions. We finally exhibit some examples of bounded domains whose squeezing
functions can be given explicitly.
