I will describe recent results from an ongoing project which examines the robustness of Kitaev's surface codes, and related FTQC schemes (due to Raussendorf and coworkers) to loss errors. The key insight is that, in a topologically ordered system, the quantum information is encoded in delocalized degrees of freedom that can be "deformed" to avoid missing physical qubits. This allows one to relate error correction and fault tolerance thresholds to percolation thresholds. Furthermore, stabilizer operators can be deformed in a similar way, which means that surface codes retain their robustness to arbitrary types of error, even when significant numbers of qubits are lost. We present numerical evidence, utilizing these insights, to show that: (1) the surface code can tolerate up to 50 percent loss errors, and (2) Raussendorf's FTQC scheme can tolerate up 25 percent loss errors. The numerics indicate both schemes retain good performance when loss and computational errors are simultaneously present. Finally we will describe extensions to other error models, in particular the case where logic gates can fail but in a heralded manner.