Abstract: The blowup-blowdown conjecture asks if a birational map between algebraic varieties can be factored into a sequence of elementary maps -- blowups and blowdowns. In the case of toric varieties the conjecture can be stated in a combinatorial form. The strong version of this conjecture is still open in all dimensions above 2. In this talk I will explain the algorithms that conjecturally solve the factorisation problem and discuss the combinatorics involved.