This is the abstract of the paper "Existence Families and Differential Operators" by Jin Liang and Ti-Jun Xiao. For the whole paper send the command get evolve-l 99-00047 to: "listserv@rz.uni-karlsruhe.de". Instead you may get the paper at "http://ma1serv.mathematik.uni-karlsruhe.de/evolve-l/index.html" too. Abstract: Let $X$, $Y$ be Banach spaces. We introduce an existence family of bounded linear operators from $Y$ to $X$, as a new tool for handling the abstract Cauchy problem in $X$: $u'(t)=Au(t)$ $(t\geq 0)$, $u(0)=x$. It turns out to be a more useful tool than the traditional operator families. We also take a look at a class of variable coefficient elliptic differential operators in $L^p(\Omega)$ ($1< p< \infty $), and reveal their relationship to our new operator system.