The increasing dominance of renewable energy sources in the modern power grids leads to the installation of power electronic inverters, as these are required to interface these DC and variable frequency sources to the AC grid. These inverters can provide additional flexibility for the control of the power grid and are distinguished into grid-forming (GFM) and grid-following (GFL) types. However, their complex control structures give rise to new threats to the stability of the system. This article investigates the transient stability of these two types of inverters. They are studied when connected to an infinite bus and subject to large disturbances, such as short-circuit faults and line disconnections. Analysis is performed for the various inverter configurations and proceeds to derive the swing equations of each configuration. The analysis is validated using time-domain simulations. Stability is then studied for a range of operating conditions and control settings. It is revealed that GFM inverters suffer greater impacts from faults in strong grid conditions, which contrasts with GFL that suffer larger impacts from faults in weak grid conditions. It is demonstrated that the virtual inertia and damping coefficients of the inverters play an important role in their transient stability and ability to ride-through faults.