In the last fifteen years or so there has been a wealth of results and methods dealing with small differences between primes (in what follows \(p\) with or without subscripts denotes primes). The author of this impressive overview paper is one of the mathematicians who has made some of the most important contributions in this field. The paper centers around the twin prime conjecture, which in its basic form states that there are infinitely many \(p\) such that \(p+2\) is prime. In the most optimistic form it states that \[ \pi_2(x) := \#\left\{p\le x:p+ 2 \text{ prime}\right\} = C\int_2^x\frac{dt}{\log^2t}+ O_\varepsilon(x^{1/2+\varepsilon}),\tag{1} \] where \[ C := 2\prod_{p>2}\left(1 - \frac{1}{(p-1)^2}\right). \] Although proving proving the basic conjecture and (1) seems out of reach nowadays, there are several ways to approximate to the twin prime conjecture. One of them is J.-r. Chen’s result [Sci. Sin. 16, 157–176 (1973; Zbl 0319.10056)] that there are infinitely many \(p\) such that \(p+2\) has at most two prime factors. Y. Zhang in 2013 [Ann. Math. (2) 179, No. 3, 1121–1174 (2014; Zbl 1290.11128)] proved that there are infinitely many pairs \((p_1, p_2)\) of distinct primes such that \[ |p_1 - p_2| \le A, \; A = 70\,000\,000.\tag{2} \] His arguments are built on the work of Goldston, Pintz and Yıldırım (so-called GPY method) [D. A. Goldston et al., Ann. Math. (2) 170, No. 2, 819–862 (2009; Zbl 1207.11096)]. They were optimized and improved by the author [Ann. Math. (2) 181, No. 1, 383–413 (2015; Zbl 1306.11073)], and the current record in (2) is \(A = 246\) [D. H. J. Polymath, Res. Math. Sci. 1, Paper No. 12, 83 p. (2014; Zbl 1365.11110)].
A related topic is the Chowla conjecture, whose weak form states that
\[ \sum_{n\le x}\lambda(n)\lambda(n+2) = o(x)\qquad(x\to\infty).\tag{3} \]
Here the Liouville function \(\lambda(n)\) equals \((-1)^n\) if \(n\) has an odd number of prime factors, otherwise \(\lambda(n)=1\). Although it was believed that (3) is as hard as the twin prime conjecture, T. Tao recently [Forum Math. Pi 4, Article ID e8, 36 p. (2016; Zbl 1383.11116)] proved a slightly weaker version of (3), namely \[ \sum_{n\le x}\frac{\lambda(n)\lambda(n+2)}{n} = o(\log x) \qquad(x\to \infty).\tag{4} \] Moreover, K. Matomäki and M. Radziwi{łł} [Ann. Math. (2) 183, No. 3, 1015–1056 (2016; Zbl 1339.11084)] proved that, for almost all intervals \([x, x+h] \subseteq [Y, 2Y]\), one has \[ \sum_{n\in[x,x+h]}\lambda(n) = O\left(\frac{h}{(\log h)^{1/10})}\right). \] The bulk of the paper consists of providing ideas and proofs behind the GPY method, sieve methods and the proof of (4). The interested reader will find seventy-five references for further study.