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Path planning is a crucial algorithmic approach for designing robot behaviors. Sampling-based approaches, like rapidly exploring random trees (RRTs) or probabilistic roadmaps, are prominent algorithmic solutions for path planning problems. Despite its exponential convergence rate, RRT can only find suboptimal paths. On the other hand, $\textrm{RRT}^*$, a widely-used extension to RRT, guarantees probabilistic completeness for finding optimal paths but suffers in practice from slow convergence in complex environments. Furthermore, real-world robotic environments are often partially observable or with poorly described dynamics, casting the application of $\textrm{RRT}^*$ in complex tasks suboptimal. This paper studies a novel algorithmic formulation of the popular Monte-Carlo tree search (MCTS) algorithm for robot path planning. Notably, we study Monte-Carlo Path Planning (MCPP) by analyzing and proving, on the one part, its exponential convergence rate to the optimal path in fully observable Markov decision processes (MDPs), and on the other part, its probabilistic completeness for finding feasible paths in partially observable MDPs (POMDPs) assuming limited distance observability (proof sketch). Our algorithmic contribution allows us to employ recently proposed variants of MCTS with different exploration strategies for robot path planning. Our experimental evaluations in simulated 2D and 3D environments with a 7 degrees of freedom (DOF) manipulator, as well as in a real-world robot path planning task, demonstrate the superiority of MCPP in POMDP tasks.