Tensors are multi-way generalizations of matrices, and similarly to matrices, they can also be factorized, that is, represented (approximately) as a product of factors. These factors are typically either all matrices or a mixture of matrices and tensors. With the widespread adoption of matrix factorization techniques in data mining, also tensor factorizations have started to gain attention. In this paper we study the Boolean tensor factorizations. We assume that the data is binary multi-way data, and we want to factorize it to binary factors using Boolean arithmetic (i.e. defining that 1+1=1). Boolean tensor factorizations are, therefore, natural generalization of the Boolean matrix factorizations. We will study the theory of Boolean tensor factorizations and show that at least some of the benefits Boolean matrix factorizations have over normal matrix factorizations carry over to the tensor data. We will also present algorithms for Boolean variations of CP and Tucker decompositions, the two most-common types of tensor factorizations. With experimentation done with synthetic and real-world data, we show that Boolean tensor factorizations are a viable alternative when the data is naturally binary.