One dimensional difference equations are widely used in population biology.
These seemingly simple models can show a variety of behaviors from stability to chaos.
 (see Cull, Yorke, May, Feigenbaum)  
We show how the enveloping technique can be used to demonstrate global and semi-global stability.
We discuss the issue of whether local stability implies global stability. 
We give some examples of more complicated behavior which can co-exist with local stability.
We show that local stability implies global stability even for models slightly more complicated than the usual models. 
We address the issue of how complicated a model must be to have local without global stability, 
and we describe our candidates for the simplest such models.