Just be honest; if you do not understand the paper, please write the editors and decline. There is nothing wrong with saying that you do not understand the article or indicating that this is not an area you are comfortable with reviewing. Don't attempt to impress the editor because the editor will never remember you.

I would like to suggest that you perform three readings of a paper, concentrating on a different element each time. At every point, remember to classify your comments as major or minor flaws. Major flaws will need considerable time to explain or correct.

I suggest that you read the very good - and brand new - book on how to write technical papers, as well as on how to assess them. The link below takes you the new edition.

https://www.maths.manchester.ac.uk/~higham/hwms/

Many publishers have materials to guide the review process. I suggest you review some of them, for example:

• https://www.springer.com/gp/authors-editors/authorandreviewertutorials/howtopeerreview
• https://authorservices.wiley.com/Reviewers/index.html

Although each publisher has its particularities, the review process is similar.

The hardest part of being a reviewer is, of course, that you need to be very well updated on the topics that are current, classical, as well as emerging. That is something that might take years to grasp, and all the while the topics move on in various directions, as well as grow - so this is not the easiest task that you can imagine.

On the other hand, you simply MUST volunteer to take on refereeing assignments, if you know that the manuscript you have received lies in your discipline - otherwise no papers can be published!

I have done 400+ peer reviews of papers, and plenty of reviews of published papers in Math Reviews, and that this is something that is as important as to publish - we need to make sure that the connections that we know are there do not become disconnected. One action is to make sure that every paper that is relevant here and now for the review is mentioned. That way we can make sure that we have a fair overview of our field.

Just be honest; if you do not understand the paper, please write the editors and decline. There is nothing wrong with saying that you do not understand the article or indicating that this is not an area you are comfortable with reviewing. Don't attempt to impress the editor because the editor will never remember you.

Ismael Joseph: This is just my personal opinion based on long-time experience: You should accept invitation to review manuscripts of other researchers only if you are already a proven expert in your field of research. This means that you must have a good overview of the field and know which results are exciting and which others are routine. Moreover, you should have published a number of peer-reviewed papers in your area of research yourself. My personal feeling is that for a graduate student it is much more important to publish own results than to get involved in peer-reviewing.

I helped my local boss in 1989 - he was also the principal of the university, in fact - with two manuscripts that he had been asked to review, and he gave them to me. That was not more than a Month or so into my PhD studies, so I knew very little. But with my supervisor's help, I grasped what to do. In fact I became a pretty good evaluator rather quickly, and have by now done roughly 400 of them!

Dear Ismael Joseph,

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