In this dissertation we study the dynamic and static probabilistic structure of the distribution of equity transaction times on financial markets. We propose dynamic, non-linear, non-Gaussian state space models to investigate both the structure of the associated inter-trade durations, and the properties of the number of transactions over a mesh of fixed length. The economic motivation of the study lies in the relationship between the properties of transaction times and those of the time-varying volatility of equity returns and of market liquidity measures such as bid-ask spreads. We use high-frequency data extracted from the Trade and Quotes database to recover transaction time-stamps recorded down to the second or millisecond time scale depending on the sample of analysis. We focus our attention to a randomly selected sub-sample of the S&P100 index traded on U.S. financial markets. Starting from the work of Chen et al. (2013), we propose a dynamic duration model that is able to capture the salient features of the empirical distribution of inter-trade durations for the most recent samples, namely, over-dispersion, long-memory, transaction clustering and simultaneous trading. We employ this model to study the structural change in the properties of the transaction process by assessing its ability of fitting the data and its forecasting accuracy over a long span of time (1993-2013). As an alternative tool for the analysis of the transaction times process, and motivated by the necessity of reducing the computational burdens induced by the appearance of data-sets of unprecedented size, we propose a dynamic, long-memory model for the number of transactions over a mesh of fixed length, based on the Markov Switching Multifractal model proposed by Calvet and Fisher (2008). We perform goodness-of-fit and forecasting accuracy comparisons against competing models and find that the proposed model provides a superior performance.