We develop analytical methodology for pricing and hedging options on the realized variance under the Heston stochastic variance model (1993) augmented with jumps in asset returns and variance. By employing a generalized Fourier transform we obtain analytical solutions (up to numerical inversion of the Fourier integral) for swaps on the realized volatility and variance and for options on these swaps. We also extend our framework for pricing forward start options on the realized variance and volatility, including options on the VIX (the Chicago Board Options Exchange Volatility Index). Our methodology allows us to consistently unify pricing and risk management of different volatility options. We provide an example of model parameter estimation using both time series of the VIX and the VIX options data and find that the proposed model is in agreement with both historical and implied market data. Finally, we derive a lognormal approximation to the density of the realized variance in the Heston model and obtain accurate approximate solutions for volatility options with longer maturities.