Abstract: Kernel density estimation (KDE) is a popular nonparametric density estimation method. We (1) derive finite-sample high-probability density estimation bounds for multivariate KDE under mild density assumptions which hold uniformly in $x \in \mathbb{R}^d$ and bandwidth matrices. We apply these results to (2) mode, (3) density level set, and (4) class probability estimation and attain optimal rates up to logarithmic factors. We then (5) provide an extension of our results under the manifold hypothesis. Finally, we (6) give uniform convergence results for local intrinsic dimension estimation.