In many biomedical studies, recurrent events are frequently encountered where subjects experience an event repeatedly over time such as multiple infections after bone marrow transplant. The gap times between these recurrent events are often the natural outcome of interest. A number of studies have been conducted to describe the gap time distribution and examine the relationship between gap times and covariates. Despite rich literature, existing methods for recurrent gap time data commonly assume that all events are of the same type and thus that the gap times are identically distributed. In various cases, however, it is inappropriate to naively adopt this assumption. In this dissertation, we study two motivating datasets, the post-transplant infection data and the South Verona psychiatric case register (PCR) data, to which existing methods cannot be directly applied. We develop nonparametric and semiparametric methods to properly analyze these recurrent gap time data. In the post-transplant infection data, enrollment to the study is triggered by transplant, which is a different event than the recurrent infections. Applying existing methods to this data leads to incorrect inferential results because the time from transplant to the first infection and the gap times between successive infections may have different distributions. We propose a nonparametric estimator of the joint distribution of the first event time and the gap times between consecutive infections. Then, a semiparametric regression method is proposed to identify risk factors for infections, accounting for the potentially different distributions of the two types of time. Often times, recurrent events consist of two alternating states. The South Verona PCR dataset is a typical bivariate alternating recurrent event dataset, in which health care and break periods alternate during follow-up. Existing methods can analyze the gap times between consecutive contacts with health care services, but the break time is ignored, which wastes useful information carried by the duration of each state. We propose a semiparametric regression method for estimating simultaneously the relationship between covariates and the durations of the two alternating states.