We study asymptotics of forward-start option prices and the forward implied volatility smile using the theory of sharp large deviations (and refinements). In Chapter 1 we give some intuition and insight into forward volatility and provide motivation for the study of forward smile asymptotics. We numerically analyse no-arbitrage bounds for the forward smile given calibration to the marginal distributions using (martingale) optimal transport theory. Furthermore, we derive several representations of forward-start option prices, analyse various measure-change symmetries and explore asymptotics of the forward smile for small and large forward-start dates. In Chapter 2 we derive a general closed-form expansion formula (including large-maturity and `diagonal' small-maturity asymptotics) for the forward smile in a large class of models including the Heston and Schobel-Zhu stochastic volatility models and time-changed exponential Levy models. In Chapter 3 we prove that the out-of-the-money small-maturity forward smile explodes in the Heston model and a separate model-independent analysis shows that the at-the-money small-maturity limit is well defined for any Ito diffusion.Chapter 4 provides a full characterisation of the large-maturity forward smile in the Heston model. Although the leading-order decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms depends highly on the parameters, and different powers of the maturity come into play. Classical (Ito diffusions) stochastic volatility models are not able to capture the steepness of small-maturity (spot) implied volatility smiles. Models with jumps, exhibiting small-maturity exploding smiles, have historically been proposed as an alternative. A recent breakthrough was made by Gatheral, Jaisson and Rosenbaum, who proposed to replace the Brownian driver of the instantaneous volatility by a short-memory fractional Brownian motion, which is able to capture the short-maturity steepness while preserving path continuity. In Chapter 5 we suggest a different route, randomising the Black-Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential Levy models and fractional stochastic volatility models. As a by-product, we make a conjecture on the small-maturity forward smile asymptotics of stochastic volatility models, in exact agreement with the results in Chapter 3 for Heston.