This thesis considers change-point and threshold problems in time series and can be separated into three principle parts. For the first part, I investigate the asymptotic inference for the threshold autoregressive (TAR) model with a structural change and establish the convergence rates and limiting distributions of the estimated thresholds and change-points. I also provide approximation method and likelihood-ratio method to make statistical inferences for the estimated thresholds and change-points. For the second part, I develop some tests for TAR models and smooth transition autoregressive (STAR) models by a separate family of hypothesis approach. I establish the limiting distributions of the test statistics under each null model and simulation results show that our approach works quite well. For the third part, I propose a threshold single-index model and construct a supremum test statistic to distinguish the threshold single-index model and the usual single-index one. A bootstrap method is used to tackle the bias problem and provide the critical values for the limiting distributions. In all the three parts, I provide extensive simulation results as well as real data examples to illustrate our methodologies and theories. The most important contributions in the thesis are to develop new asymptotic theories when the threshold and changepoint both exist, and to establish tests to distinguish some threshold-type time series models. These results have made quite a big step forward on the theories of change-point and threshold time series models.