The following report deals with some completion problems of partial matrices; specifically, we shall analyze totally nonnegative partial matrices, totally nonpositive partial matrices, R and TR-partial matrices.
The aim is to know the current status of these problems and provide necessary and sufficient conditions to enable us to close several open cases.

In the first part, we introduce the essential concepts to understand and handle the problems under study.
Additionally, we also show the tools used throughout this work, with emphasis on Graph Theory, which plays an important role in the analysis of partial matrices since partial matrices can be represented by directed or undirected graphs.

The second part is devoted to the totally nonnegative completion problem.
After a presentation of the current state of the problem, we analyze the case of matrices with the main diagonal that are partially specified. The main results, however, we will obtain when the main diagonal is to be specified.
We will generalize some known results for the case of positionally symmetric partial matrices and obtain new results, in some cases closing problems, for partial matrices that are not positionally symmetric.
The last chapter also presents some of the issues that are still open.

Finally, we dedicate the third part of this report to analyze the tota\-lly nonpositive completion problem, and
the R-matrices as well as the TR-matrices completion problem. With the achieved results, we managed to resolve the last two mentioned problems.