In this paper we present a preliminary research report on the problem of nding procedures to solve some cases of the Uniform E-Semi- unication Problem, where E is a set of equations dening an equational theory. The problem in general is undecidable, but for certain theories, as is the case with E-Unication, there may be theories for which it is decidable. 1 Introduction The Semi-Unication problem has applications in areas of computer science like programming languages, computational linguistics, term rewriting, and automated deduction. Despite this, the body of knowledge on Semi-unication is not comparable to that of Unication, which has been for a long time an ongoing branch of study in computer science. The general problem of semi-unication (see below for denitions) has been shown to be undecidable, but there are a number of restrictions andor generalizations that can be shown to be decidable. In this paper we present an introduction and a preliminary research rep...