In this paper, we establish the first large deviation bounds for the Airy point process. The proof is based on a novel approach which relies upon the approximation of the Airy point process using the Gaussian unitary ensemble (GUE) up to an exponentially small probability, together with precise estimates for the stochastic Airy operator and edge rigidity for beta ensembles. As a by-product of our estimates for the Airy point process, we significantly improve upon previous results on the lower tail probability of the one-point distribution of the KPZ equation with narrow-wedge initial data and the half-space KPZ equation with Neumann boundary parameter $A=-\frac{1}{2}$ and narrow-wedge initial data in a unified and much shorter manner. Our bounds hold for all sufficiently large time $T$, and for the first time establish sharp super-exponential decay with exponent $3$ for tail depth less than $T^{\frac{2}{3}}$ (with sharp leading prefactors $\frac{1}{12}$ and $\frac{1}{24}$ for tail depth less than $T^{\frac{1}{6}}$).