Much is known about simple rings and these results are often extended to direct sums of simple rings or to a ring, R, whose factor ring R/I is simple for a suitable ideal I. An alternative approach to extending these results is to loosen the condition simplicity puts on ideals of a ring. Just infinite dimensional algebras are those which, in some sense, contain only few ideals. I'll start by defining just infinite more precisely. My talk will summarize known results for and examples of just infinite rings. I'll also discuss the results of my dissertation- a few properties of just infinite algebras and an example of broadening theorems about simple rings to theorems about just infinite rings.