This paper introduces a family of "generalized long-memory time series models", in which observations have a specified conditional distribution, given a latent Gaussian fractionally integrated autoregressive moving average (ARFIMA) process. The observations may have discrete or continuous distributions (or a mixture of both). The family includes existing models such as ARFIMA models themselves, long-memory stochastic volatility models, long-memory censored Gaussian models, and others. Although the family of models is flexible, the latent long-memory process poses problems for analysis. Therefore we introduce a Markov chain Monte Carlo sampling algorithm and develop a set of recursions which make it feasible. This makes it possible, among other things, to carry out exact likelihood-based analysis of a wide range of non-Gaussian long-memory models without resorting to the use of likelihood approximations. The procedure also yields predictive distributions that take into account model parameter uncertainty. The approach is demonstrated in two case studies.