We deal with the qualitative behaviour of the first-passage-time density  
of a one-dimensional diffusion process X(t) over a moving boundary; 
in particular, we study  the value that the  first-passage time density takes 
at  zero, the distribution of the maximum process, and the distribution 
of the first instant at which X(t) attains the maximum 
in an interval [0,T]. Our results generalize the analogous ones already 
known for Brownian motion. Some examples are reported.