In this talk we will provide a method for obtaining examples of domains (both Euclidean and non-Euclidean) satisfying Polya's inequality, by relating this to the second term in the Weyl asymptotics. On the other hand, we will also provide examples showing that the nature of this second term and the truth of Polya's conjecture are independent. If time allows, we will show how to improve current known bounds in the case of Euclidean cylinders of the form IxA, where I is a bounded interval and A a bounded n-dimensional Euclidean domain.