  The explicit criteria for several types of ergodicity of one-dimensional
diffusions or birth-death processes have been found out recently in a
surprisingly short period. One of the criteria is for exponential ergodicity of
birth-death processes. This problem has been opened for a long time in the
study of Markov chains. The survey article explains in details the idea which
leads to solve the problem just mentioned. It is interesting that the problem
is connected with several branches of mathematics. Some open problems for the
further study are also proposed.
