The origin of life requires the emergence of a system of autocatalytic polymers such as RNA. We consider a trans-acting replicase that catalyses replication of a template (either a copy of itself or another sequence). Our model includes alternating plus/minus strand replication where only the plus strand is a catalyst. Prebiotic chemistry generates random sequences and allows for non-catalysed, template-directed synthesis of new strands. These chemical reactions are insufficient to sustain replication, but they provide a background in which the first replicase can arise. In the well-mixed case, the minimum value of the catalytic rate parameter k for which a stable replicating state survives scales as 1/f, where f is the fraction of random sequences that are catalysts. When catalysts are rare (f→0), the replicating state is not stable in for any finite k because the replicases are overrun by parasitic templates already present in the prebiotic system, and by additional parasites created by mutation of the catalyst. In contrast, in 2d spatial simulations, the replicating state is stable for moderate k with appropriate values of the local diffusion constant. We calculate the probability of spread of the replicating state from a single isolated catalyst. This occurs in a parameter range that is narrower than that in which existing replicators are stable. The 2d model uses 'Two׳s Company' rules, where two molecules on a site may replicate, but crowding occurs when three molecules are on one site. A mean-field theory is presented which predicts the most important results of the spatial model. Our results emphasize that the origin of replication is a spatially-localized stochastic transition between a 'dead' state controlled by prebiotic chemistry and a 'living' state controlled by autocatalytic replication.