An ongoing project centred around Robin Pemantle (www.cs.auckland.ac.nz/~mcw/Research/mvGF) is aimed at deriving asymptotic approximations of multiply-indexed sequences via analysis of the set $V$ of singularities of the associated multivariate generating function $F$. In many commonly occurring cases of interest to this audience, the general shape of the asymptotics is well understood and the leading term is known fairly explicitly in terms of the geometry of $V$. I will give a brief overview of the project and then concentrate on recent work aimed at the efficient explicit computation of higher order terms in such expansions. This includes combinatorial applications and numerical results. One possible application of such work is to extend the class of $F$ which we can deal with to include algebraic functions, and I will also discuss this briefly.