We introduce noncausal processes to the count time series literature. These processes are defined by time-reversing an INAR(1) process, a non-INAR(1) Markov affine count process, or a random coefficient INAR(1) [RCINAR(1)] process. In the special cases of INAR(1) and RCINAR(1), the causal process and its noncausal counterpart are closely related through a same queuing system with different stochastic specifications. The noncausal processes we introduce are generically time irreversible and have some unique calendar time dynamic properties that are unreplicable by existing causal models. In particular they allow for locally bubble-like explosion, while at the same time remaining stationary. These processes have closed form calendar time conditional probability mass function, which facilitates nonlinear forecasting.