Halacha Corner: Is Running Permitted on Shabbat?
Running deep dive: When is it allowed on the day of rest?

The book of Isaiah (chapter 58) states: "If you turn away your foot from Shabbat, from doing your own pleasure on My holy day, and call Shabbat a delight, the holy day of Hashem honorable, and shall honor Him, not doing your own ways, nor finding your own pleasure, nor speaking your own words."
From the statement "honor Him, not doing your own ways," our sages in Tractate Shabbat (page 113) learned: "Your walking on Shabbat should not be the same as your walking on weekdays." Therefore, a person should not run on Shabbat, whether light running or fast running, as ruled by the Rif, the Rosh, the Tur, and Maran in the Shulchan Aruch.
Running for a Mitzvah
In the Talmud, Tractate Berachot (page 6): "Rabbi Zeira said, initially when I saw Torah scholars running to study Torah in the study hall on Shabbat, I would say they were desecrating Shabbat, until I heard Rabbi Tanchum say in the name of Rabbi Yehoshua ben Levi, one should always run for a matter of halacha even on Shabbat, as it says (in Hosea chapter 11) 'They shall walk after Hashem, who roars like a lion.'" From here we learn that it is permitted to run on Shabbat for the purpose of a mitzvah.
Running for Pleasure
The Smag wrote that young men who enjoy running and jumping are permitted to run on Shabbat, as they are not running for business purposes, and furthermore, they enjoy their running. This was also ruled by the Tur and Maran in the Shulchan Aruch.
Similarly, a person walking outside when it starts to rain, or who is concerned that it might rain, is permitted to run, as it is evident that he is not running for business purposes, and it is also his pleasure to run, because if he refrains from running he will be distressed by getting rained on. It is also permitted to run away from a dog that is chasing him and in similar situations.
If one is walking and encounters a puddle of water, he may jump over it, even if it is so wide that he cannot place his first foot down before lifting the second (meaning both feet are in the air simultaneously, which is normally forbidden as it resembles running on Shabbat). It is better to jump over it than to walk around it, because walking around would cause greater effort.
Summary:
Running is forbidden on Shabbat. However, when running brings pleasure to a person and there is no concern that it appears as if one is going about their business affairs, such as children and young people who enjoy running, or a person fleeing from rain and similar situations, it is permitted for all these to run on Shabbat.
Rulings of Rabbi Ovadia Yosef courtesy of "Daily Halacha" website