  In the Multiagent Path Finding problem (MAPF for short), we focus on
efficiently finding non-colliding paths for a set of $k$ agents on a given
graph $G$, where each agent seeks a path from its source vertex to a target. An
important measure of the quality of the solution is the length of the proposed
schedule $\ell$, that is, the length of a longest path (including the waiting
time). In this work, we propose a systematic study under the parameterized
complexity framework. The hardness results we provide align with many
heuristics used for this problem, whose running time could potentially be
improved based on our fixed-parameter tractability results.
  We show that MAPF is W[1]-hard with respect to $k$ (even if $k$ is combined
with the maximum degree of the input graph). The problem remains NP-hard in
planar graphs even if the maximum degree and the makespan$\ell$ are fixed
constants. On the positive side, we show an FPT algorithm for $k+\ell$.
  As we delve further, the structure of~$G$ comes into play. We give an FPT
algorithm for parameter $k$ plus the diameter of the graph~$G$. The MAPF
problem is W[1]-hard for cliquewidth of $G$ plus $\ell$ while it is FPT for
treewidth of $G$ plus $\ell$.
