to run; to hurry (when going somewhere)
to drive (a car) quickly; to ride fast (on a horse)
to win (fame)
故郷に思いを馳せた。
I let my thoughts run to my hometown.
名を馳せている武将は多い。
There are many commanders who have made a name.
遠い異国の地に想像を馳せる楽しみがある。
There's a pleasure in letting your imagination run to distant foreign lands.