The overall topic of this thesis is approximate martingale estimating function-based estimationfor solutions of stochastic differential equations, sampled at high frequency. Focuslies on the asymptotic properties of the estimators. The first part of the thesis deals with diffusions observed over a fixed time interval. Rate optimal and effcient estimators areobtained for a one-dimensional diffusion parameter. Stable convergence in distribution isused to achieve a practically applicable Gaussian limit distribution for suitably normalisedestimators. In a simulation example, the limit distributions of an effcient and an ineffcientestimator are compared graphically. The second part of the thesis concerns diffusions withfinite-activity jumps, observed over an increasing interval with terminal sampling time goingto infinity. Asymptotic distribution results are derived for consistent estimators of ageneral multidimensional parameter. Conditions for rate optimality and effciency of estimatorsof drift-jump and diffusion parameters are given in some special cases. Theseconditions are found to extend the pre-existing conditions applicable to continuous diffusions,and impose much stronger requirements on the estimating functions in the presenceof jumps. Certain implications of these conditions are discussed, as is a heuristic notion ofhow effcient estimating functions might be constructed, thus setting the stage for furtherresearch.