The Annual Review is a pretty good magazine for review articles.
The Concept of the Anthropocene (Malhi, 2017) is written by a proponent of the concept of the anthropocene.
At its core, the Anthropocene is an encapsulation of the concept that modern human activity is large relative to the basic processes of planetary functioning, and therefore that human social, economic, and political decisions have become entangled in a web of planetary feedbacks.
Or, "Actually, the earth is small, and the ocean is not an infinite garbage dump."
How it started
The modern use of the term Anthropocene began in 2000 with Crutzen & Stoermer's paper in the Global Change Newsletter, simply entitled “The Anthropocene.” This was followed in 2002 by Crutzen's high-profile piece in Nature (“Geology of Mankind”), which gained much wider circulation and attention.
Crutzen is a 1995 Nobel Laureate for research on the ozone hole.
A pivotal event in terms of gaining wider scientific acceptance and adoption was the publication of a thematic issue of Philosophical Transactions of the Royal Society in 2011. This issue covered a range of perspectives, including conceptual and historical antecedents (6), biosphere transformation (10), sediment fluxes (11), and the geological case (12).
That issue is
13 March 2011, Volume 369, Issue 1938. It's a very fun issue, plenty to read!