The L-curve is a popular heuristic to tune Tikhonov regularization in linear inverse problems. This paper shows how it naturally arises when the problem is solved from a Bayesian perspective. Specifically, it establishes that the L-curve is a graphical way of searching for the maximum a posteriori solution after marginalization over the priors. The framework is general enough to enclose the classical (linear, logarithmic and square-root) L-curves as particular cases and to allow the design of new L-curves. It also explicitly accounts for the dimensions of the inverse problem (number of observations versus number of unknowns) in regularization. Elaborating on this framework, new criteria for locating the corner of the L-curve are discovered, such as the ‘minimum speed on the curve’ and the ‘maximum angular speed’, and conditions are established for their equivalence with the maximum curvature and the marginalized maximum a posteriori . All results are supported by numerical experiments. Experiments also show that the Bayesian L-curve rooted on appropriate priors can succeed in inverse problems where the classical L-curve is prone to fail.