In the context of climate change, emissions of different species (e.g., carbon dioxide and methane) are not directly comparable since they have different radiative efficiencies and lifetimes. Since comparisons via detailed climate models are computationally expensive and complex, emission metrics were developed to allow a simple and straightforward comparison of the estimated climate impacts of emissions of different species. Emission metrics are not unique and variety of different emission metrics has been proposed, with key choices being the climate impacts and time horizon to use for comparisons. In this paper, we present analytical expressions and describe how to calculate common emission metrics for different species. We include the climate metrics radiative forcing, integrated radiative forcing, temperature change and integrated temperature change in both absolute form and normalised to a reference gas. We consider pulse emissions, sustained emissions and emission scenarios. The species are separated into three types: CO2 which has a complex decay over time, species with a simple exponential decay, and ozone precursors (NOx, CO, VOC) which indirectly effect climate via various chemical interactions. We also discuss deriving Impulse Response Functions, radiative efficiency, regional dependencies, consistency within and between metrics and uncertainties. We perform various applications to highlight key applications of emission metrics, which show that emissions of CO2 are important regardless of what metric and time horizon is used, but that the importance of short lived climate forcers varies greatly depending on the metric choices made. Further, the ranking of countries by emissions changes very little with different metrics despite large differences in metric values, except for the shortest time horizons (GWP20).