In this project we will investigate several open problems in the theory of Beurling generalized primes. We focus on the following topics: Well-behaved number systems. Such systems have very strong error terms in the asymptotic laws for their prime and integer counting functions. We will investigate existence questions for such systems. Malliavin's problem. This problem concerns the determination of the best possible error term in the abstract prime number theorem in the case of an integer law with Malliavin-type remainder. This is a long-standing open problem, and the best result currently known is still far off the conjectured form. We aim to improve this record. The Möbius function. For Beurling primes it is not known in general if the prime number theorem implies that the Möbius function has a mean value (necessarily equal to zero). Our goal is to settle this question. Furthermore we will investigate possible generalizations of the Selberg-Delange method, a well-established technique from classical analytic number theory, to a general Tauberian theorem. We will also investigate applications to Beurling generalized primes.