I have undertaken an overview of research papers about Open Access (OA) to answer the question as to where are we now on the road to fully open scientific publishing. I hope that this summary will be as helpful to people who are generally familiar with the idea of Open Access, as to those who frequently take part in discussions on its current state.
Open Access means free, unrestricted access to scholarly publications, which became possible mostly thanks to the Internet. At this moment, it is technically possible to obtain any kind of digital content from almost anywhere in the world. Maintaining on-line, digital content is much cheaper than printing and distributing books or journals, and this led to the idea of publishing scientific works free of charge to readers.
Green and Gold Open Access
The biggest problem with gold Open Access is that high-quality Open Access journals are not (yet) present in every field of science. Although this is changing, both through the conversion of traditional journals to Open Access and by establishing new, fully OA publications. You can find a list of Open Access journals, ordered by disciplines, publishers, country, etc. in the Directory of Open Access Journals. What is more, there are also a growing number of gold Open Access publishing opportunities for book authors (have a look here).
Green OA is limited by publishers policies and agreements that are signed by authors. Although at this moment self-archiving of Open Access copies is allowed for the large majority of journals  (you can find information on the policy of each particular journal here). GGreen OA still also lacks the necessary infrastructure. Most authors place copies of their work on their personal websites or research groups websites , which often disappear, thus green Open Access is very often a temporary Open Access. Repositories are a much better way to store articles for a longer time, even though some of them struggle to achieve visibility (have a look here).Moreover, a study has shown that only 37% of authors know of a suitable subject repository. 
Open Access share
IIt is very hard to determinate the exact share of Open Access in the publishing market. The studies conducted on Web of Knowledge and Scopus databases have shown that over 20% of articles published in academic journals are freely available on the Internet and it is probable that both green and gold options have similar shares of the market [1, see also a chart]. Classical work by Mikael Laakso and Bo-Christer Björk claimed that OA journal publishing has steadily increased its relative share of all scholarly journal articles by about one percent annually over the 2001-2011 decade .
Image source: 
There is big variation between the disciplines in popularity of both green and gold open access models. It is well known, that green Open Access is very popular in physics, mathematics and earth scientists, but also among researchers in the fields of social sciences, arts and humanities who tend to republish free copies of their articles.  The interesting thing is that green OA is less popular in the life and medical sciences. However, there is a large number of well known fully Open Access journals in these fields.  It looks like green and gold roads are alternative, not complimentary means to OA.
Open Access publishers
The majority of OA journals are published by university presses and scientific associations, and one third are published by professional publishers. The domination of non-commercial publishers is clearly seen in the fields of social sciences (including business and economics), arts and humanities, while it is less so in medicine, biology, mathematics, engineering and physics.  In short, one may say that commercial publishers operate mostly in the fields with bigger funding.
A large number of journals published by societies and universities are established, previously conventional journals, converted into an OA model, in contrast to journals run by commercial publishers, almost 80% of which were started in Open Access  (most of them in the 21st century). The emergence of these new journals, published by commercial publishers, most notably in the fields of biological and medical studies, was one of the most important changes in the history of Open Access. Newcomers were able achieve rapid recognition, and a number of them now make up the top tier of OA publishing.
Image Source: 
Open Access business models
Commercial publishers usually cover the costs of publishing Open Access journals with Article Processing Charges (APC), paid for by authors or institutions that support authors. APCs are also used by other kinds of publishers, but not as often. The average APC is about 900$ per article , but this depends significantly on the discipline and the country of journal location. The highest fees are charged by medical journals.
Image source: 
What is worth noting is that money paid as APCs comes in the final count from the very same source as subscription fees, since both subscriptions and APCs are paid by research institutions to cover costs of publishing books and journals by academic publishers. The difference is that in the subscription-based model the money makes academic knowledge available only to the staff of a single institution, and in the case of Open Access it allows access to everyone in the world. This made possible projects such Sponsoring Consortium for Open Access Publishing in Particle Physics (SCOAP3), which is an association of research institutions working in the field of high-energy physics. In cooperation with numerous publishers (both commercial and non-commercial) from different countries, the association was able to convert 13 out of 15 journals in this field to Open Access. Similar organizations may play an important role in the further development of Open Access.
Source of the first image: 
I have used Open Access copies of following articles:
1. Björk, Bo‐Christer, et al. “Anatomy of green open access.” Journal of the Association for Information Science and Technology 65.2 (2014): 237-250.
2. Laakso, Mikael, and Bo-Christer Björk. “Anatomy of open access publishing: a study of longitudinal development and internal structure.” BMC medicine 10.1 (2012): 124.
3. Solomon, David. “Types of open access publishers in Scopus.” Publications 1.1 (2013): 16-26.
4. Solomon, David and Bo‐Christer Björk. “A study of open access journals using article processing charges.” Journal of the American Society for Information Science and Technology 63.8 (2012): 1485-1495.