"This dissertation consists of two essays dealing respectively with estimation of volatility and test for a jump using high frequency data. Chapter 1 investigates the properties of pre-averaging estimators of integrated volatility, first considered by Podolskij and Vetter (2009). We relax their assumptions on the properties of market microstructure noise in order to include realistic and empirically relevant features of noise such as missing data and flat price trading. We develop an asymptotic theory of our estimator using martingale convergence theorems. Especially we deal with the boundary problem of pre-averaging and we provide a solution to the parameters-on-the-boundary problem posed by pre-averaging estimators. Building on that theory, we show that a general linear combination of estimators can be made unbiased, and we devise a rate-optimal estimator of the integrated volatility. In addition, we derive a bootstrap statistic to assess the variance of our estimator. This allows us to optimally select the estimator's smoothing parameter from the data, providing an additional improvement over previously-considered pre-averaging estimators. Because our methodology and assumptions on the market microstructure noise component are general, our estimator can also be applied to multivariate time series without any need to correct for asynchronicity in the observations. Monte Carlo experiments show that our theoretical results are valid in realistic cases. Chapter 2 shows that the power of any test of this hypothesis depends on the frequency of observation. In particular, we show that if the process is observed at intervals of length 1/n and the instantaneous volatility of the process is given by [sigma]t, at best one can detect jumps of height no smaller than [sigma]t[...characters removed]. We construct a test which achieves this rate in the case for diffusion-type processes. With simulation experiments, we show that our tests have good size and power properties in many cases with realistic sample sizes and that they outperform other tests that have been proposed in the recent literature. Applying our tests to high-frequency financial data, we detect more jumps in the data than are found by other tests."--Leaves v-vi.