In this paper we introduce the space-fractional Poisson process whose state probabilities $p_k^α(t)$, $t>0$, $α\in (0,1]$, are governed by the equations $(\mathrm d/\mathrm dt)p_k(t) = -λ^α(1-B)p_k^α(t)$, where $(1-B)^α$ is the fractional difference operator found in the study of time series analysis. We explicitly obtain the distributions $p_k^α(t)$, the probability generating functions $G_α(u,t)$, which are also expressed as distributions of the minimum of i.i.d.\ uniform random variables. The comparison with the time-fractional Poisson process is investigated and finally, we arrive at the more general space-time fractional Poisson process of which we give the explicit distribution.