We present two results on the periodic points of one-dimensional maps. The 
first result is a sufficient condition for the coexistence of $ 2^n$-periodic points
for general, not necessarily unimodal, one dimensional maps. Our condition is
simple and easy to check.  Similar sufficient conditons are also given for the
existence of 2- and 3-periodic points. The second is concerned with the period
doubling bifurcations of one-parameter family of unimodal maps and establishes
the global existence of bifurcation branches of $ 2^n $ -periodic points for 
all $n = 0,1,2, \cdots$. This extends the local result given by Collet, Eckmann
and Lanford III [1].