In a manifold with connection, the starting point of a martingale depends on its final law, and also on the final random variable and the underlying filtration. Different notions of convexity will be investigated, guaranteeing existence and uniqueness of martingales with prescribed terminal value. I will start with results from the 90' and finish with very recent results on martingales with reflection and possible drift on simply connected open sets of the plane, in relation with reflected backward stochastic differential equations.