We propose Gamma conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class $A_F$ to a Fano manifold $F$. We say that $F$ satisfies Gamma conjecture I if $A_F$ equals the Gamma class ${\hat{\Gamma }}_F$. When the quantum cohomology of $F$ is semisimple, we say that $F$ satisfies Gamma conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by ${\hat{\Gamma }}_{F}Ch\left(E_i\right)$ for an exceptional collection $\left\{E_i\right\}$ in the derived category of coherent sheaves ${\mathcal{D}}_{\mathrm{coh}}^b\left(F\right)$. Gamma conjecture II refines a part of a conjecture by Dubrovin. We prove Gamma conjectures for projective spaces and Grassmannians.