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We present analytical results for the distribution of first-passage (FP) times of random walks (RWs) on random regular graphs that consist of $N$ nodes of degree $c \ge 3$. Starting from a random initial node at time $t=0$, at each time step $t \ge 1$ an RW hops into a random neighbor of its previous node. In some of the time steps the RW may hop into a yet-unvisited node while in other time steps it may revisit a node that has already been visited before. We calculate the distribution $P( T_{\rm FP} = t )$ of first-passage times from a random initial node $i$ to a random target node $j$, where $j \ne i$. We distinguish between FP trajectories whose backbone follows the shortest path (SPATH) from the initial node $i$ to the target node $j$ and FP trajectories whose backbone does not follow the shortest path ($\lnot {\rm SPATH}$). More precisely, the SPATH trajectories from the initial node $i$ to the target node $j$ are defined as trajectories in which the subnetwork that consists of the nodes and edges along the trajectory is a tree network. Moreover, the shortest path between $i$ and $j$ on this subnetwork is the same as in the whole network. The SPATH scenario is probable mainly when the length $\ell_{ij}$ of the shortest path between the initial node $i$ and the target node $j$ is small. The analytical results are found to be in very good agreement with the results obtained from computer simulations.