  Let $X$ be a smooth projective Fano variety over the complex numbers. We
study the moduli spaces of rational curves on $X$ using the perspective of
Manin's Conjecture. In particular, we bound the dimension and number of
components of spaces of rational curves on $X$. We propose a Geometric Manin's
Conjecture predicting the growth rate of a counting function associated to the
irreducible components of these moduli spaces.
