TITLE: How to write a good MathSciNet review?
QUESTION [73 upvotes]: When reviewing for MathSciNet, I routinely find myself just paraphrasing and abbreviating the introduction provided by the author, and occasionally adding a few words about the quality of the research or the cleverness of the argument (which the authors themselves would not be able to write for obvious reasons). There seems to be very little added value in doing this, since the paper already has the introduction (which has the added benefit of being written by someone who has intimate knowledge of the paper) as well as abstract (which will in most cases be sufficient to decide if the paper is worth reading). Of course, I can imagine edge cases when someone can't quite decide if the paper is worth delving into based on the abstract alone, while the introduction is for some reason difficult to read or the paper is difficult to access. But I can't help feeling that there should be more to it.
I would love to hear opinions about what makes a MathSciNet review useful, and how to achieve it.
Edit to add: As YCor correctly points out, the same question applies with ZBMath or any other place that hosts public reviews in place of MathSciNet. To avoid creating a question which is a moving target, I will refrain from making edits above.

REPLY [23 votes]: I'm the Executive Editor at Mathematical Reviews.  I'm impressed by some of the answers, particularly those from Timothy Chow and Denis Serre, which give as good an answer as I could hope for. As Timothy Chow and Joe Silverman point out, having links to other items in MathSciNet related to the current paper is especially helpful to users.  Kimball quotes the salient bits of our Guide for Reviewers.  Thank you for doing that. People tend to forget that the Guide exists.  When I talk with people in person at a conference, I compress the goal of a good review down to: describe something about the context of the result (where did it come from?); describe the main result (what is it?); describe the main technique(s) or method(s) (how did they do it?).  Doing all this concisely can be difficult, of course.
The AMS Bulletin still publishes reprints of reviews.  They are generally tied to one of the articles in the issue.  While they are generally above average, they are not necessarily meant to be the best of the best.  For a time, I was posting particularly good reviews that I came across.  You can find them via this link: exceptional reviews. I should probably get back to posting examples of good reviews.  I second Yemon Choi's recommendation for Kimball's collection of (his version of) exceptional reviews.