This is a special Analysis/Princeton-IAS Number Theory seminar.  Goldston & Montgomery and Montgomery & Soundararajan have established formulae for the variance of sums of the von Magoldt function over short intervals (i.e. for the variance of the number of primes in these intervals) assuming, respectively, the pair-correlation conjecture and the Hardy-Littlewood conjecture. I will discuss the generalisation of these formulae to other arithmetic functions associated with the Selberg class of L-functions, in the context of both zero statistics and arithmetic correlations. I also hope to discuss the function-field analogues of these generalisations.