  We develop the theory of Abelian functions associated with algebraic curves.
The growth in computer power and an advancement of efficient symbolic
computation techniques has allowed for recent progress in this area. In this
paper we focus on the genus three cases, comparing the two canonical classes of
hyperelliptic and trigonal curves. We present new addition formulae, derive
bases for the spaces of Abelian functions and discuss the differential
equations such functions satisfy.
