Bilevel optimization has garnered growing interest over the past decade. However, little attention has been paid to detecting and dealing with unboundedness in these problems, with most research assuming a bounded high-point relaxation. In this paper, we address unboundedness in bilevel optimization by studying its computational complexity and developing algorithmic approaches to detect it. We show that deciding whether an optimistic linear bilevel problem is unbounded is strongly NP-complete. Furthermore, we extend the theoretical intractability result to the multilevel case, by showing that for each extra level added, the decision problem of checking unboundedness moves up a level in the polynomial hierarchy. Finally, we introduce two algorithmic approaches to determine whether a linear bilevel problem is unbounded and, if so, return a certificate of unboundedness. This certificate consists of a direction of unboundedness and corresponding bilevel feasible point. We present a short proof of concept of these algorithmic approaches on some relevant examples.