In prevalent cohort survival studies where subjects are recruited at a cross-section and followed prospectively in time, the observed event times are length-biased and further follow a multiplicative censoring scheme. For such studies there is an associated initiation time which may be unknown. In this case we only observe the time from sampling to the event of interest. This is the forward recurrence time. Further in such cases standard left-truncation survival analysis methods are not applicable. In other scenarios like current duration studies, the time of the initiating event may be known but there is no subsequent follow-up after sampling. Here we observe the backward recurrence times. In presence of covariates, the proportional hazards model may not be applicable to forward and backward recurrence time data. However, due to the invariance of the accelerated failure time model under length bias and cross-sectional sampling, it can serve as a useful alternative. In particular, existing estimators for the regression parameter like the ordinary least squares and Tsiatis’ log rank estimators may be valid. The problem however is that these estimators are based on the conditional distribution of the time variable given the covariates. Under length bias sampling, the covariate distribution is functionally dependent on the regression parameter. Thus a ``naive'' analysis conditioning on the covariates may result in information loss. We show that if the covariate distribution is left completely unspecified then there is no loss of information under a conditional analysis. We also perform simulation studies to compare our method to the existing methods for forward and backward recurrence time data analysis. Finally, we analyze time-to-pregnancy data comparing our method to ordinary least squares regression. Next, we show the connection between k-monotone densities and forward and backward recurrence time data. We show that if we start with a k-monotone density, the corresponding recurrence time density is (k+1)-monotone. So, to use k-monotone density estimation for forward recurrence time data, we develop an algorithm for consistent estimation of a k-monotone density under right censoring. We determine the rate of convergence and asymptotic distribution of the proposed estimator. We look at the viability of the estimator under some simulation settings and also apply it to the ARIC data. Finally, we look at the effect of recurrence time on competing risks. We determine the recurrence time subdistributions and also develop an algorithm for estimating the original subdistributions. We show the consistency and determine the asymptotic distribution of the proposed estimator. We look at simulation results to determine the efficacy of the estimator and also a data application for the ARIC data.