This paper presents some recent time series models (the so-% called NAREX(1) models) for Laplace variables with first order autoregressive structures. They are analogs of standard AR(1) model and of the LAR(1), NLAR(1) and AREX(1) models, introduced by Andel, Lawrance, Lewis, Jevremović and others. Some of their models can be obtained from NAREX models as special cases. The distribution of the innovation sequence (a probability mixture) and the autoregressive structure of NAREX processes are discussed as well.