  Motivated in part by various sequences of graphs growing under random rules
(like internet models), convergent sequences of dense graphs and their limits
were introduced by Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi and by
Lov\'asz and Szegedy. In this paper we use this framework to study one of the
motivating class of examples, namely randomly growing graphs. We prove the
(almost sure) convergence of several such randomly growing graph sequences, and
determine their limit. The analysis is not always straightforward: in some
cases the cut distance from a limit object can be directly estimated, in other
case densities of subgraphs can be shown to converge.
