The defining trait of planeswalkers is the ability to travel between separate universes with ease, while the vast majority of people throughout the multiverse are not even aware that other worlds exist besides their own. Planeswalking is a form of magic. With enough time and mana, or with specialized spell knowledge, or with access to enormous power, it's possible for a planeswalker to transfer clothing, artifacts, and/or creatures with them as they planeswalk.
A planeswalker is specifically a being who possesses a planeswalker's spark. The planeswalker spark is more or less a one-in-a-million thing in sentient beings, and having it ignite is even rarer. There are other beings who, through various means, are able to travel between planes, but they are not considered planeswalkers (Marit Lage, the Eldrazi, and the Myojin of Night's Reach are the best-known examples). Many prerevisionist characters were referred to as planeswalkers but may not actually have been; without any further information, they remain subject to debate.
Planeswalkers can be born at random in any sapient species, with no outward signs of their latent power. However, there is an incredibly remote chance that any given sentient and natural being will be born with a planeswalker's spark. When that being is put through a period of extreme stress—in many cases death—the spark can trigger, causing the individual to ascend and become a planeswalker.
The act of Planeswalking is a spell, and like how every mage practices magic differently, every planeswalker planeswalks slightly differently. Some can planeswalk swiftly, some more slowly. Some require great effort to planeswalk, some can do it with ease. Some can planeswalk again in a short amount of time, some can take a while. Some can carry more inorganic material with them. A rare few can planeswalk organic material (and those usually have restrictions - Yanggu, for example, can only planeswalk with Mowu specifically, and Wrenn needs to be in a simbiotic relation with a Treefolk to take them with her). Kaya is exceptional in that she can merge her body and mind with another being and take that individual with her on a planeswalk. Nissa had a packet of Zendikari seeds she planned to plant on another plane. Ugin was able to bring Nicol Bolas to the Meditation Plane wrapped in his wings, though the Blind Eternities grievously injured and blinded Bolas, and it is implied that Bolas only survived the trip because he was a powerful Elder Dragon.
Planeswalkers can planeswalk to specific locations. A planeswalker can effectively teleport by planeswalking to another plane and then back.
Chandra describes her sense of other planes and planeswalking as akin to her peripheral senses.
Any planeswalker can follow another planeswalker in their immediate wake.
For a creature to become a planeswalker under normal circumstances, they must have a soul and be a mage of some kind.  Artificial creatures like Angels and Demons, which are manifestations of mana, do not have the right soul or essence to hold a spark, although Calix managed to turn into a Planeswalker through some unknown process.
Planeswalkers had incredible magical capabilities, surpassing all but the most powerful mortal wizards. Their lives could last indefinitely, and their physical forms were matters of will as they were energy projections of a center of consciousness. Through intense effort, planeswalkers could create their own artificial planes. Because of planeswalkers' prolonged life spans and immense power, some are worshipped as gods; many end up losing their sanity, or, at the very least, they come to regard the lives of mortals in low-esteem, if even at all.
The new breed of planeswalkers no longer displays the near-omnipotence of their predecessors. While they are usually powerful mages, they are still physical beings that in general age normally, can be harmed, and need the same sustenance as other mortals. They stopped being able to transport other people during a planeswalk. They can bring their clothes and some small items but, for example, not food. This is in stark contrast to the earlier planeswalkers. Some of them have managed to suppress or avoid some of these limitations by magical means; however, these are specific to each planeswalker.
The new breed manifested itself for the first time in Venser of Urborg, a Dominarian artificer who participated in the solution of the Dominarian temporal crisis. Teferi's first theory was that the rifts mutated Venser's spark, which affected his ascension.
The new breed was born during the Mending, when Jeska sacrificed her life and her spark to mend all temporal rifts in the Multiverse (doing so on such a great scale was probably enabled by her former existence as Karona, the embodiment of Dominarian magic, and the fact that Dominaria is the Nexus of the Multiverse). The Mending caused a change in the very rules of the Multiverse and in the nature of the planeswalker sparks.
Reasons for change
Pivotal for the Mending was the creative team's long-standing wish to make planeswalkers more identifiable. Toning them down provided a solution that also cleared the ways for the new planeswalker card type. This, in turn, allowed planeswalkers to be not only the focus of the storyline but also of brand identity.
A further change was made in Commander 2014: planeswalker cards no longer represent the full power of planeswalkers themselves, but only the amount of assistance a planeswalker is willing to provide to the player. Such changes made any planeswalker, including pre-Mending planeswalkers, available for printing as a planeswalker card in the future.
As with most changes, the reactions were mixed. Some deemed it unnecessary to kill off existing characters, arguing that they could have been altered to fit the new approach. Others felt that diminishing their powers made the characters less interesting. Additional criticism was directed at the way the Mending was handled in the Time Spiral Cycle. An open letter was written to Brady Dommermuth that summarizes these viewpoints on Phyrexia.com.
Discussions on differences between the old and new planeswalkers spawned many (sometimes malicious) names for the latter type, generally to make them easier to refer to, but also to show how much they differ from the original ones. Among the most popular are "neowalkers", from Greek neos ("new"), and "Bradywalkers", named after Brady Dommermuth, creative director. For the same (non-malicious) reason, the original breed of planeswalkers is often referred to as "oldwalkers."
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