Let S be a string built on some alphabet Σ. A multi-cut rearrangement of S is a string S′ obtained from S by an operation called k-cut rearrangement, that consists in (1) cutting S at a given number k of places in S, making S the concatenated string X1⋅X2⋅X3…Xk⋅Xk+1, where X1 and Xk+1 are possibly empty, and (2) rearranging the Xis so as to obtain S′=Xπ(1)⋅Xπ(2)⋅Xπ(3)…Xπ(k+1), π being a permutation on 1,2…k+1 satisfying π(1)=1 and π(k+1)=k+1. Given two strings S and T built on the same multiset of characters from Σ, the Sorting by Multi-cut Rearrangements (SMCR) problem asks whether a given number ℓ of k-cut rearrangements suffices to transform S into T. The SMCR problem generalizes and thus encompasses several classical genomic rearrangements problems, such as Sorting by Transpositions and Sorting by Block Interchanges. It may also model chromoanagenesis, a recently discovered phenomenon consisting in massive simultaneous rearrangements. In this paper, we study the SMCR problem from an algorithmic complexity viewpoint, and provide, depending on the respective values of k and ℓ, polynomial-time algorithms as well as NP-hardness, FPT-algorithms, W[1]-hardness and approximation results, either in the general case or when S and T are permutations.