  We study the spaces of rational curves on Fano threefolds with Gorenstein
terminal singularities. We generalize the results regarding Geometric Manin's
Conjecture for smooth Fano threefolds, including the classification of
subvarieties with higher a-invariants and Movable Bend-and-Break lemma. We also
show Geometric Manin's Conjecture for some singular del Pezzo threefolds.
