In this work, we developed a general approach to the problem of detecting and quantifying different types of correlations in bipartite quantum systems. Our method is based on the use of distances between quantum states and processes. We rely upon the premise that total correlations can be separated into classical and quantum contributions due to their different nature. In addition, according to recently discussed criteria, we determined the requirements to be satisfied by distances in order to generate correlation measures physically well behaved. The proposed measures allow us to quantify quantum, classical and total correlations. Besides the well-known case of relative entropy, we introduce some additional examples of distances which can be used to build bona fide quantifiers of correlations.