Gemara

(Rav Nachman): If one was walking and does not know where the Techum ends, he takes 2000 medium steps. This is the Techum.

Berachos 6b (R. Zeira): At first, when I saw Talmidim running to the Kalah (public lecture before the festival), I thought that they were disgracing Shabbos;

Later, I heard R. Tanchum teach that one should run to learn (some texts - to do a Mitzvah), even on Shabbos, for it says "Acharei Hash-m Yelchu k'Aryeh Yish'ag." Now, also I run.

Shabbos 113a (Beraisa): "Me'Asos Derachecha" - your Halichah (going or walking) on Shabbos should not be like your Halichah during the week.

113b - Question: How should one walk differently on Shabbos?

Answer #1: Rav Huna taught that if one encounters a stream, if he can leave one foot on the ground and step across, it is permitted [even to jump]. If not, he may not [jump across].

Objection (Rava): Going around causes extra walking! Rather, since there is no alternative, he may jump across.

Answer #2 (R. Yosi b'Rebbi Yehudah): One may not take a big step on Shabbos, and not even during the week. This detracts from one's vision. It is restored at Kidush of Shabbos night.

Rishonim

Rambam (Hilchos Shabbos 24:4): One may not run or jump on Shabbos. "Me'Asos Derachecha" teaches that your going (walking) on Shabbos may not be like your going during the week.

Tur (OC 301, also brought in Hagahos Maimoniyos Hilchos Shabbos 24:2): Sefer ha'Mitzvos says that youths who enjoy jumping and running are permitted [on Shabbos]. Similarly, to see anything they enjoy, it is permitted to see it.

Beis Yosef (DH Kasuv): The words 'it is permitted' at the end of the Tur are extra. They are not in Semak. He already wrote 'it is permitted'!

Bach (DH Aval): I say that the extra words at the end of Semak teach a difference between the cases. Jumping on Shabbos is permitted only b'Di'eved, since they are used to running and jumping for enjoyment on Shabbos. If we would protest, we will deprive them of Oneg Shabbos. L'Chatchilah, if they ask, we do not permit them. We suspect that during the week they are used to doing so for profit. The Semak says 'youths who enjoy running and jumping', i.e. he already feels that this is Oneg for him, even though he does not profit from it, so it is permitted for him, like his habit. He repeats "permitted to see", to teach that not only may one see something he already saw on Shabbos and knows that he enjoys it. Rather, even something he never saw, but he thinks that it will be Oneg for him, l'Chatchilah he may run or jump to see it. We need not suspect that it is for profit. For youths who are used to it, it is a Safek whether they do so in order to profit.

Taz (1): I say that the Tur added these words, lest one think that the Heter to run and jump is in order to go see something he enjoys. Really, he forbids this. This is like the Isur to exercise. Surely this is when he gets some enjoyment from exercising, e.g. it arouses appetite. We do not discuss fools who run for no reason! Even so, it is forbidden, for he has no enjoyment while running. We do not permit due to the Oneg that will come later. This is unlike youths who enjoy jumping and running at the time they jump and run. Therefore, one may not jump and run in order to see something. Rather, the Tur permits seeing alone. One might have thought that seeing is forbidden. The Tur (Sof Siman 306) permits thinking about one's business, but one should not do so at all, due to Oneg Shabbos. The Beis Yosef brings from R. Yonah that we are concerned lest amidst thinking, he come to be distracted. We pray for "... Menuchah Sheleimah sha'Atah Rotzeh Bah." One might have thought that Hash-m does not want us to see and think about superficial matters. The Tur teaches that it is permitted, for this is Oneg Shabbos.

Bi'ur Halachah (DH Kol): Chemed Moshe explains that the extra words in the Tur come to exclude looking at something forbidden to see, e.g. theaters. Surely we do not permit an Isur for Oneg Shabbos!

Rashi (42a DH Pesi'ah): An average step of a person is one Amah.

Poskim

Shulchan Aruch (OC 301:1): One may not run on Shabbos, unless it is to do a Mitzvah, e.g. to the Beis ha'Keneses or a similar matter.

Rema: One may not take a step bigger than one Amah, if he can take a smaller step.

Gra (DH v'Asur): Eruvin 42a teaches that a normal step is one Amah.

Magen Avraham (2): The Agudah brings from Tosfos that one should not remove a foot from the ground before the other foot rests on the ground.

Mishnah Berurah (2): Likewise, one may not jump on Shabbos.

Shulchan Aruch (2): Youths who enjoy jumping and running are permitted.

Magen Avraham (3): The Shulchan Aruch connotes that this is l'Chatchilah, unlike the Bach.

Gra: The Tosefta forbids running in order to exercise. This shows that it is forbidden only if it is for exercise that is not enjoyable.

Mishnah Berurah (5): It is permitted, for this is their Oneg.

Kaf ha'Chayim (9): This refers to youths playing together.

Kaf ha'Chayim (11): One may dance on Simchas Torah, or at Simchas Nisu'in for the honor of Chasan and Kalah.

Shulchan Aruch (ibid.): And similarly [it is permitted] to see anything they enjoy seeing.

Bach (3): Semak forbids running and jumping in order to profit, but permits for a Mitzvah. Youths who enjoy running and jumping may do so, since it is not for profit. The same applies to seeing something he enjoys. I.e. one may run or jump in order to see it. They fulfill a Mitzvah through running and jumping, since it is not to profit. That is surely forbidden, since it is the way one goes during the week.

Mishnah Berurah (6): The Shulchan Aruch permits running in order to see something he enjoys.

Magen Avraham (4): Even though one may not see illustrations (307:15), it is permitted if he enjoys it.

Eshel Avraham: The Shulchan Aruch forbids reading captions under illustrations. The Magen Avraham explains that if he has Oneg and enjoys it only if he understands what is illustrated, he may read the caption.

Kaf ha'Chayim (12): The Magen Avraham holds like the Taz, but the Acharonim hold like the Bach, who permits running and jumping in order to see something he enjoys.

Rema: Similarly, one may stroll.

Beis Yosef (DH uv'Tosefta): A Tosefta (Shabbos 17:16) forbids running for exercise. One may stroll normally, even the entire day.

Beis Yosef (DH Kasuv bi'Teshuvos): A Teshuvah at the end of Sefer Chazeh ha'Tenufah says that strolling on Shabbos in a Mavoy with an Eruv, with the head uncovered, depends on the local custom. It is good if the custom is not to, and [then] one may not be lenient. Pesachim 51a proves so [that if people are stringent about something, one may not be lenient in front of them].

Kaf ha'Chayim (14, citing Machazik Berachah 2): Chazeh ha'Tenufah is an abridged version of Teshuvos ha'Rosh. It says there (Siman 55) "if the custom is to stroll, this is a good custom. One should not be lenient, for it is a proper custom, and to cover oneself differently on Shabbos than during the week. The custom of our ancestors is Torah."

Magen Avraham (5): One may stroll even if he intends for exercise and to warm himself. One may not run in order to warm himself. This requires investigation, for he enjoys this! We must say that it is forbidden when he does so for Refu'ah.

Mishnah Berurah (7): It is permitted even if he intends for exercise and to warm himself for Refu'ah, since it is not evident that it is for Refu'ah. One may not run in order to warm himself for Refu'ah, since it is evident. It is forbidden due to [a decree against all Refu'os] lest one grind ingredients. Some are stringent even about strolling if he intends to exercise for Refu'ah.