We consider general stochastic volatility models with no local volatility component and derive the general expression of the volatility smile at order two in volatility-of-volatility. We show how, at this order, the smile only depends on three dimensionless numbers whose precise expressions as functionals of the model's spot/variance and variance/variance covariance functions we provide.Finally we assess the accuracy of our order two expansion using realistic levels of volatility-of-volatility.