You may get an email from a journal or conference claiming to be legitimate, with known names on its masthead, offering you the chance to publish your work or present for a small fee. If it feels fishy, investigate further! Once your work is published anywhere, even in an illegitimate journal or conference proceeding, other publishers will not want to accept your article.
Predatory publishers will promise quick turnaround times but the articles they publish are poorly peer-reviewed or not at all, nor are the publications considered reliable by other researchers or promotion-and-tenure committees.
A 2019 symposium of publishing societies, research funders, researchers, policymakers, academic institutions, libraries and others in the biomedical field developed the following definition to fully name the issue that had emerged in the 21st century (Grudniewicz et al., 2019):
Predatory journals and publishers are entities that prioritize self-interest at the expense of scholarship and are characterized by false or misleading information, deviation from best editorial and publication practices, a lack of transparency, and/or the use of aggressive and indiscriminate solicitation practices.
This symposium was guided by the Principles of Transparency and Best Practice in Scholarly Publishing (linked below).
The Predatory Paradox: Ethics, Politics, and Practices in Contemporary Scholarly Publishing by 
Visit the publisher website and look for the following (adapted from Grudniewicz et al., 2019 and Elmore & Weston, 2020):
Do an internet search for the publisher and editorial board.
While there is no single, exhaustive list of trustworthy publishers, these tools can help you gather information as you come to your own determination.
These journal directories include journals judged to be trustworthy after having met a set of criteria based on editorial practices and other factors.
Retraction Watch hosts a volunteer-led effort to list journals proven to have been "hijacked", pretending to be legitimate. Does the journal you're wondering about appear on this list?
Consider looking up the website on the WhoIs Domain Lookup. Hijacked journals often use domains that look very similar to those of legitimate journals, but the domain registration information can show if the URL is actually registered to the journal publisher or not.
Checklists to review when making a decision.
This guide aims to help researchers identify and avoid predatory publishers, but it should not be considered legal advice. The final responsibility for any publishing decision lies with the researcher themselves.
