Free open-access peer-reviewed math journals Is there any free (as in free beer, i.e., no publication fees or other fees whatsoever), open-access (free and open access to everyone) and peer-reviewed mathematics journal?
I am interested in a list of journals. I am particularly interested in my field which is low dimensional topology and geometry and geometric group theory.
The observation how well a site like MathOverflow or Math Stack Exchange works makes me wonder if a similar system does exist for the peer-reviewing and publication process itself. 
 A: Australasian Journal of Combinatorics: http://ajc.maths.uq.edu.au
Moved a couple of years ago from print/subscription to online/free, using the accumulated surplus from many years of small profits on subscriptions to fund the ongoing costs (which are not very high). 
A: There are two excellent (new) open access journals from Cambridge University Press:
Forum of Mathematics, Sigma: http://journals.cambridge.org/action/displayJournal?jid=FMS
and
Forum of Mathematics, Pi: http://journals.cambridge.org/action/displayJournal?jid=FMP
I think they both cover the spectrum of pure mathematics (though they don't say 'pure' in their descriptions)
A: International Journal of Group Theory is a free peer-reviewed journal published by the University of Isfahan in English.
A: The original question asked about low dimensional topology and geometry. For sufficiently algorithmic research in these areas, the Journal of Computational Geometry may be relevant. For instance, a recent issue includes a paper on the space complexity of recognizing low-dimensional manifolds.
Another algorithmic journal that I regularly publish in, free-as-in-beer to both authors and readers, is the Journal of Graph Algorithms and Applications.
A: Two more FAPP (for all practical purposes). Nobody mentioned them.
With slighty more applied emphasis http://www.complex-systems.com/contribute.html
All issues besides the current one (four per year) are open access and there are no fees.
Also, don't forget PNAS. http://www.pnas.org/site/authors/guidelines.xhtml Open access after one year. Very fast reviewing. PNAS has always published top quality short mathematical papers. http://www.pnas.org/content/by/section/Mathematics?FIRSTINDEX=10
A: Annales Academiæ Scientiarum Fennicæ is a very good, free and open access journal. It primarily publishes papers in analysis but is open to all fields of mathematics. Geometric group theory fits pretty well into the profile of the journal. The MathSciNet Mathematical Citation Quotient for 2016 was 0.75.
A: Annales Mathematicae Silesianae
Annales Mathematicae Silesianae publishes significant research and survey papers from all branches of pure and applied mathematics, and reports of meetings. It welcomes contributed papers that develop important, new mathematical ideas and results or solve outstanding problems.
The journal does not have article processing charges (APCs) nor article submission charges
A: The Far Eastern Mathematical Journal is free access and peer-reviewed journal publishing in Russian and English.
A: http://www.compositionality-journal.org/

Compositionality describes and quantifies how complex things can be assembled out of simpler parts. Compositionality, the journal, is an open-access journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline. Topics may concern foundational structures, an organizing principle, or a powerful tool. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition.

A: Combinatorial Theory is a new open-access journal in combinatorics that replaces Journal of Combinatorial Theory, Series A, whose entire editorial board recently resigned to from it (effective January 2021).
Algebraic Combinatorics is the result of a similar effort, and is the successor of Journal of Algebraic Combinatorics, a Springer journal that continues in diminished form.
A: You're asking for "diamond" open access (to contrast with the gold open access --- peer reviewed but not free --- and the green open access --- free but not peer reviewed). Here are some math journals of this type, that use arXiv as a repository (socalled "overlay journals"):


*

*Discrete Analysis (blog)

*Symmetry, Integrability and Geometry

*Logical Methods in Computer Science
A more complete list can be accessed at the Directory of Open Access Journals (doaj.org)
A: Aside from Theory and Applications of Categories, which I mentioned in the comments, there is a new journal that publishes higher category theory and related areas, namely Higher Structures. 

This journal publishes articles that make significant new contributions to mathematical science using higher structures, or that significantly advance our understanding of the foundational aspects of the theory of such structures. The scope of the journal includes: higher categories, operads and their generalisations, and applications of these to Algebra, Geometry, Topology, Combinatorics, Logic and Mathematical Physics.

One very good point is that the journal uses an explicit open access license, rather than merely being free (as in your favourite bubbly drink) to read.
A: *

*Cahier de topologie et géométrie différentielle catégoriques

*Theory and Applications of Categories

*Categories and General Algebraic Structures with Applications

*New York Journal of Mathematics
A: Last year I compiled a list of such journals related that are either generic or related to algebraic geometry: http://math.commelin.net/agdoaj.html.
For a long list of journals see https://thomas1111.wordpress.com/2013/06/29/some-serious-open-acess-journals/ by Thomas Sauvaget.
A: Advances in Combinatorics is a new arXiv overlay journal that ''aims to be a top-level specialist journal in combinatorics.'' The journal has a very strong editorial board with Dan Král' and Tim Gowers serving as managing editors.  See this post and this post on Tim Gowers's blog for more information.  As of now, the journal is completely free beer, with financial and administrative support from Queen's University Library.
A: The Bulletin of the Irish Mathematical Society
A: As other answers mention, such journals certainly exist. Two examples of general journals that fit this definition are
Journal de l'École polytechnique — Mathématiques
Documenta Mathematica
A: The New York Journal of Mathematics (http://nyjm.albany.edu/) is a peer reviewed and free online general math journal. It has been publishing since the mid-1990s. It is not an ArXiv overlay journal. Published articles appear on the journal website.
A: Acta Mathematica, a top journal, is fully open access by special arrangement with the Institut Mittag-Leffler.
A: Yes; only in my research area there are two: ETNA (Electronic Transactions on Numerical Analysis) and ELA (Electronic Journal of Linear Algebra). They are both from noncommercial publishers, one funded by a university and one by a professional society.
A: Annales de l'Institut Fourier are now an open access journal: http://aif.cedram.org
Some other journals in France (serious but less known):
Confluentes Mathematici: http://cml.cedram.org
North-Western European Journal of Mathematics: http://math.univ-lille1.fr/~nwejm/
One can also mention the Electronic Journal of Differential Equations
http://ejde.math.txstate.edu/ which exists since 1993
A: The Journal of Singularities. The editorial board is impressive, and it seems to be pretty much an ideal case of open access journals: "... In order to fund the Center, and to expand our activities in the future, we rely on video-textbook sales (at low prices), licensing deals with institutions and publishers, donations, investors, and advertising revenue. Our goal is to grow to the point where we can fund additional activities, such as: in-office tutorials and review sessions for undergraduates in the Greater Boston area, 24/7 video-chat tutoring, postdoctoral research positions (including office space), hosting of mathematics research conferences, and adding more Worldwide journals."
A: Journal of Logic and Analysis
"This journal examines the interaction between ideas or techniques from mathematical logic and other areas of mathematics, especially, but not limited to, pure and applied analysis. "
A: The Hardy Ramanujan Journal publishes papers in number theory. It is totally open access, peer reviewed and is an epijournal based on the HAL platform.
A: There is a list of Diamond free open access journal here:
https://www.cimpa.info/en/node/62
We try to keep it updated.
A: There are plenty of journals which charge no fees, and online access is free.
I am surprised that nobody mentioned the Bulletin of the AMS.
Some other examples are Conformal Geometry and Dynamics (electronic only):
 http://www.ams.org/journals/ecgd/2014-18-06/S1088-4173-2014-00265-3/
Journal of mathematical Physics, analysis and geometry (only electronic version is free), http://jmage.ilt.kharkov.ua/
All these are peer-reviewed and actually of high quality.
A: The Bulletin of the Iranian Mathematical Society is free access and peer-reviewed journal publishing in English.
A: A very recent (diamond open-access) journal has been lauched with the support of many major research institutions in France:

Annales Henri Lebesgue

Also, the editors and founders wrote a small paper (French, English) in order to emphasize the relevance of struggling for an open access world nowadays, and describing how the journal works. 
A: INTEGERS: The Electronic Journal of Combinatorial Number Theory
Published since its inception 15+ years ago as free and peer-reviewed electronic journal; by now it is in addition available in print (not freely, of course).
In addition to Combinatorial Number Theory, as the name indicates, it also has a focus on Game Theory.
A: Journal of Integer Sequences:
https://cs.uwaterloo.ca/journals/JIS/ 
Electronic Journal of Qualitative Theory of Differential Equations
http://www.math.u-szeged.hu/ejqtde/index.html
A: The Journal of Nonsmooth Analysis and Optimization is a "fair" (in particular diamond) open access journal that launched this year on the episciences arXiv overlay platform.
Other journals on that platform that I haven't seen mentioned yet are the journal of Groups, Complexity, Cryptology and the Epijournal de Géométrie Algébrique.
(Disclaimer: I am one of the Managing Editors of JNSAO.)
A: The American Mathematical Society has a new diamond open access journal called the Communications of the American Mathematical Society.
