This work proposes a resilient and adaptive state estimation framework for robots operating in perceptually-degraded environments. The approach, called Adaptive Maximum Correntropy Criterion Kalman Filtering (AMCCKF), is inherently robust to corrupted measurements, such as those containing jumps or general non-Gaussian noise, and is able to modify filter parameters online to improve performance. Two separate methods are developed – the Variational Bayesian AMCCKF (VB-AMCCKF) and Residual AMCCKF (R-AMCCKF) – that modify the process and measurement noise models in addition to the bandwidth of the kernel function used in MCCKF based on the quality of measurements received. The two approaches differ in computational complexity and overall performance which is experimentally analyzed. The method is demonstrated in real experiments on both aerial and ground robots and is part of the solution used by the COSTAR team participating at the DARPA Subterranean Challenge.