Given a sequence X = (X 1 , X 2 , . . .) of random observations, a Bayesian forecaster aims to predict X n+1 based on (X 1 , . . . , Xn) for each n ≥ 0. To this end, she only needs to select a collection σ = (σ 0 , σ 1 , . . .), called "strategy" in what follows, where σ 0Because of the Ionescu-Tulcea theorem, σ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability is to be selected. In a nutshell, this is the non-standard approach to Bayesian predictive inference. A concise review of the latter is provided in this paper. We try to put such an approach in the right framework, to make clear a few misunderstandings, and to provide a unifying view. Some recent results are discussed as well. In addition, some new strategies are introduced and the corresponding distribution of the data sequence X is determined. The strategies concern generalized Polya urns, random change points, covariates and stationary sequences.