In the Multi-Agent Meeting problem (MAM), the task is to find the optimal meeting location for multiple agents, as well as a path for each agent to that location. Among all possible meeting locations, the optimal meeting location has the minimum cost according to a given cost function. Two cost functions are considered in this research: (1) the sum of all agents paths' costs to the meeting location (SOC) and (2) the cost of the longest path among them (MKSP). MAM has many real-life applications, such as choosing a gathering point for multiple traveling agents (humans, cars, or robots). In this paper, we divide MAM into two variants. In its basic version, MAM allows multiple agents to occupy the same location, i.e., it is conflict tolerant. For MAM, we introduce MM*, a Multi-Directional Heuristic Search algorithm, that finds the optimal meeting location under different cost functions. MM* generalizes the Meet in the Middle (MM) bidirectional search algorithm to the case of finding an optimal meeting location for multiple agents. Several admissible heuristics are proposed for MM*, and experiments demonstrate the benefits of MM*. As agents may be embodied in the world, a solution to MAM may contain conflicting paths, where more than one agent occupies the same location at the same time. The second variant of the MAM problem is called Conflict-Free Multi-Agent Meeting (CF-MAM), where the task is to find the optimal meeting location for multiple agents (as in MAM) as well as conflict-free paths (in the same manner as the prominent Multi-Agent Path Finding problem (MAPF)) to that location. For optimally solving CF-MAM, we introduce two novel algorithms, which combine MAM and MAPF solvers. We prove the optimality of both algorithms and compare them experimentally, showing the pros and cons of each algorithm.