Robot localization is an inverse problem of finding a robot's pose using a map and sensor measurements. In recent years, Invertible Neural Networks (INNs) have successfully solved ambiguous inverse problems in various fields. This paper proposes a framework that solves the localization problem with INN. We design an INN that provides implicit map representation in the forward path and localization in the inverse path. By sampling the latent space in evaluation, Local\_INN outputs robot poses with covariance, which can be used to estimate the uncertainty. We show that the localization performance of Local\_INN is on par with current methods with much lower latency. We show detailed 2D and 3D map reconstruction from Local\_INN using poses exterior to the training set. We also provide a global localization algorithm using Local\_INN to tackle the kidnapping problem.