In multi-label classification, an instance is associated with multiple relevant labels, and the goal is to predict these labels simultaneously. Many real-world applications of multi-label classification come with different performance evaluation criteria. It is thus important to design general multi-label classification methods that can flexibly take different criteria into account. Such methods tackle the problem of cost-sensitive multi-label classification (CSMLC). Most existing CSMLC methods either suffer from high computational complexity or focus on only certain specific criteria. In this work, we propose a novel CSMLC method, named progressive random k-labelsets (PRAkEL), to resolve the two issues above. The method is extended from a popular multi-label classification method, random k-labelsets, and hence inherits its efficiency. Furthermore, the proposed method can handle arbitrary example-based evaluation criteria by progressively transforming the CSMLC problem into a series of cost-sensitive multi-class classification problems. Experimental results demonstrate that PRAkEL is competitive with existing methods under the specific criteria they can optimize, and is superior under other criteria.