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In finance theory the price is often supposed to follow an Ito semimartingale while no explicit assumptions are made on the dynamic evolution of trading volumes. Trading volume is a measure of the quantity of shares that change owners for a given security. The amount of daily volume on a security can fluctuate on any given day depending on the amount of new information available about the company. We assume that the dynamic evolution of trading volume is represented as a semimartingale.  Analogously to stock prices, the stochastic process for trading volume might be characterized by diffusive and jump components. We distinguish between two classes of widely used processes: Brownian semimartingales plus jumps and pure-jump models. The relative contribution of each of two components is estimated by means of alternative nonparametric methods. We also analyze if the jump component is a stochastic process of finite or infinite variation. Finally, alternative parametric models are estimated and compared. The paper analyzes the specification of continuous-time models for high-frequency trading volume series.