We present a complete description of the small time to expiry asymptotics of the Black-Scholes implied volatility under the condition of no arbitrage and a very weak technical condition. The presented formulae are in terms of the option and stock prices, strike price and time to expiry. In the process of obtaining these results, we derive small time to expiry asymptotics of the Black-Scholes price of a European call option as well as a number of interesting properties of the time-scaled implied volatility. Our results show that close to expiry the at-the-money (ATM) implied volatility behaves very diŽerently to the implied volatility of options that are not ATM. The ¯nal section of the note demonstrates that implied volatility may fail to converge, to a ¯nite or in¯nite limit, as time to expiry goes to zero even in the simple case of the Black-Scholes model with time-dependent volatility. The talk will also give an overview of some other issues and results related to the existence and properties of implied volatility in a (local) martingale model of the stock price.