We study an online version of the robust mean change point detection problem under a dynamic Huber contamination model with arbitrary contamination distribution and inlier distribution possessing exponentially- or polynomially-decaying tails. This robustness framework is systematically studied for the first time in the change point literature. For univariate data, we characterise the detection delay by partitioning the parameter space into four regimes, in terms of the true change location, signal size and contamination level. Efficient detection procedures are accompanied by matching lower bounds, up to poly-logarithmic factors. For the multivariate setting, we devise an efficient robust mean testing procedure and apply this to the robust online change point problem. The theoretical analysis of the robust mean testing procedure is the first in dealing with both Huber contamination and heavy-tailedness, and is thus of independent interest. Extensive numerical experiments are conducted to support our theoretical findings.