What is the difference between the current-rate method and the temporal method for translating foreign operations, and when is each used?

The current-rate (closing-rate) method translates assets and liabilities at the closing rate, income and expenses at the average rate, and equity at historical rates, with the resulting differences taken to OCI (the translation reserve / CTA). The temporal (remeasurement) method translates non-monetary items at historical rates and monetary items at the closing rate, with differences going to profit or loss. Under IAS 21 the choice is driven by functional currency: when a foreign operation's functional currency differs from the group's presentation currency, you use the current-rate method to translate it for consolidation. The temporal method (remeasurement) applies when an entity keeps its books in a currency other than its own functional currency — e.g. an integral operation whose functional currency is effectively the parent's — so its results must be remeasured into the functional currency first. Hyperinflation is a separate regime: under IAS 29 you first restate the subsidiary's statements for inflation and then translate at the closing rate — you do NOT use the temporal method for it.