Auto-Ignition is a phenomenon that occurs in many practical combustion processes (engine knock, ignition in rapid compression machines or shock tubes etc). For many purposes, the combustible mixture can be treated as uniform in space, allowing zero-dimensional modelling. Sometimes, however, non-uniformities in temperature or in pressure cause “hot spot” formation, having increased temperature with respect to the surrounding. Since ignition delay is highly temperature dependent, the hot spot will ignite much earlier than its surrounding, leading to space- and time-dependent processes governed by the superposition of chemistry, gas-dynamics, and transport.This paper presents mathematical models to simulate homogeneous and hot spot ignition in one-dimensional geometries by solution of the conservation equations using detailed chemistry and a multi-species transport model. Spatial discretization of the resulting partial differential equation system on an adaptive non-uniform grid leads to a system of ordinary differential/algebraic equations that is solved numerically by implicit methods.Results are presented for homogeneous ignition processes in i-octane/n-heptane mixtures and for 1D ignition processes in the hydrogen-oxygen system. The results show that the model used is able to describe the spatial and temporal development of auto-ignition, explosions as well as detonations, that are initiated by hot spots.