Correlated survival data with possible censoring are frequently encountered in survival analysis. This includes multi center studies where subjects are clustered by clinical or other environmental factors that influence expected survival time, studies where times to several different events are monitored on each subject, and studies using groups of genetically related subjects. To analyze such data, we propose accelerated failure time (AFT) models based on lognormal frailties. AFT models provide a linear relationship between the log of the failure time and covariates that affect the expected time to failure by contracting or expanding the time scale. These models account for within cluster association by incorporating random effects with dependence structures that may be functions of unknown covariance parameters. They can be applied to right, left or interval-censored survival data. To estimate model parameters, we consider an approximate maximum likelihood estimation procedure derived from the Laplace approximation. This avoids the use of computationally intensive methods needed to evaluate the exact log-likelihood, such as MCMC methods or numerical integration that are not feasible for large data sets. Asymptotic properties of the proposed estimators are established and small sample performance is evaluated through several simulation studies. The fixed effects parameters are estimated well with little absolute bias. Asymptotic formulas tend to underestimate the standard errors for small cluster sizes. Reliable estimates depend on both the number of clusters and cluster size. The methodology is used to analyze data taken from the Minnesota Breast Cancer Family Resource to examine age-at-onset of breast cancer for women in 426 families.