In this talk I will discuss certain questions concerning the asymptotic behavior of the Epstein zeta function E_n(L,s) in the limit of large dimension n. In particular I will describe the value distribution of E_n(L,s) for a random lattice L of large dimension n, giving partial answers to questions raised by Sarnak and Strömbergsson in their study of the minima of E_n(L,s). Many of the key ingredients in our discussion will come from the rich interplay between the value distribution of the Epstein zeta function and classical problems in the geometry of numbers