Let K be a totally real number field with ring of integers O_K. Let U_K(n) be the minimal rank of an n-universal O_K-lattice; i.e. the smallest positive integer k such that there exists a rank k positive definite O_K-lattice which represents all rank n positive definite O_K-lattices. With the exception of a finite number of real quadratic fields, we prove an explicit asymptotic formula for log U_K(n) as n tends to infinity. We also show that, for any constant C > 0 and n > 2, there are only finitely many totally real fields K such that U_K(n) < C, with all such fields being effectively computable. Similarly, for any n > 2, we show that there are only finitely many totally real fields K admitting an n-universal criterion set S_K(n) of size less than C, with all such fields likewise being effectively computable. This talk is based on joint work with Dayoon Park, Pavlo Yatsyna, and Jongheun Yoon.