  We investigate the question: when is a higher-rank graph C*-algebra
approximately finite dimensional? We prove that the absence of an appropriate
higher-rank analogue of a cycle is necessary. We show that it is not in general
sufficient, but that it is sufficient for higher-rank graphs with finitely many
vertices. We give a detailed description of the structure of the C*-algebra of
a row-finite locally convex higher-rank graph with finitely many vertices. Our
results are also sufficient to establish that if the C*-algebra of a
higher-rank graph is AF, then its every ideal must be gauge-invariant. We prove
that for a higher-rank graph C*-algebra to be AF it is necessary and sufficient
for all the corners determined by vertex projections to be AF. We close with a
number of examples which illustrate why our question is so much more difficult
for higher-rank graphs than for ordinary graphs.
