  In 1988 Serrano \cite{Ser}, using Reider's method, discovered a minimal
bielliptic surface in $\PP^4$. Actually he showed that there is a unique family
of such surfaces and that they have degree 10 and sectional genus 6. In this
paper we describe, among other things, the geometry of the embedding of the
minimal bielliptic surfaces. A consequence of this description will be the
existence of smooth nonminimal bielliptic surfaces of degree 15 in $\PP^4$. We
also explain how to construct the degree 15 surfaces with the help of the
quadro-cubic Cremona transformation of $\PP^4$. Finally, we remark that the
quintic elliptic scroll and the abelian and bielliptic surfaces of degree 10
and 15 are essentially the only smooth irregular surfaces known in $\PP^4$ (all
others can be derived via finite morphisms $\PP^4\to\PP^4 $).
