Abstract: A. Beurling introduced an abstraction of prime number theory in the 1930's in which only multiplicative structure is present. A sequence of real numbers ${\cal P}: \lambda_1 \le \lambda_2 \le \cdots$, with $\lambda_1 > 1$ and $\lambda_n \to \infty$, is taken to be a set of g-primes, and the products $\lambda_1^{k_1}\lambda_2^{k_2}\cdots \lambda_r^{k_r}$ to be the associated set of g-integers. Let $N_{\cal P}(x)$ denote the number of such products not exceeding $x$ (taking account of any multiplicity) and let $\pi_{\cal P}(x) = \#\{{\cal P} \cap [1,x]\}$. The central questions concern the relations between $N_{\cal P}(x)$ and $\pi_{\cal P}(x)$. Beurling proved that if $N_ {\cal P}(x) = cx +O\big(x/(\log x)^\kappa\big)$ with $c > 0$ and $\kappa > 3/2$ then $\pi(x) \sim x/\log x$, i.~e. the analog of the PNT holds for $\cal P$. We describe some of the problems in Beurling g-number theory and recent progress, including Kahane's $L^2$ condition for the PNT and the oscillation result of Montgomery, Vorhauer, and the speaker.}{dia}