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A Twin Flame (also Twin Soul) is a spiritual (New Age) concept describing a special soul connection between two halves of the same soul. The Twin Flames are thought to be a template for a new type of relationship between lovers. The fundamental thought behind this concept is that the dawning new era in human spiritual evolution will be a time when relationships foster enhanced spiritual growth between lovers, whereas in previous times and still early in the 21st century couples stay together for purposes of physical survival and economical safety more than anything else. According to the mythology of Twin Flames, in the beginning of time we were created as a perfect soul, that was split into two soul halves - one half female, the other half male - that would then be cast upon Earth to be forever looking for one-another. They would reincarnate over lifetimes with this longing for each other, and once they finally meet, they would reunite and be in love and then leave this physical plane as one whole individual soul.
There are plenty of modernised theories of Twin Flames, many of which presenting the view that there is a twin flame for everyone (thus confusing catalytic relationships and soul mates to twin flames) and that the halves of twin flames are not necessarily a male and female, but can also be homosexual. Also, some theorists accept that this is not a relationship that can be experienced once in all life cycles only, but repeats over several. However, most take it as a sign of spiritual enlightenment and a sign of evolution. Despite these differing views Twin Flame concept is still very much tied to monogamous ideals, and the thought that only graduate souls have twin flames on Earth as their purpose is to do spiritual work and energy work to help the Planet. However, that does not mean a Twin Flame relationship would be something to aspire for, as soul mates, are much more important due to the mere higher number of soul mates doing spiritual work, and a soul mate relationship is often more harmonious; Twin flames go through a lot of cleansing and purging of the chakras and energetiacal bodies, and with true Twin Flames the connection is so intense that it causes fear and many twin flames initially run from the experience (the Twin Flame runner). 
The first mention of what has been later interpreted to be a description of Twin Flames comes from Plato:
In his dialogue The Symposium, Plato has Aristophanes present a story about soul mates. Aristophanes states that humans originally had four arms, four legs, and a single head made of two faces. He continues that there were three genders: man, woman and the "Androgynous". Each with two sets of genitalia with the Androgynous having both male and female genitalia. The men were children of the sun, the women were children of the earth and the Androgynous were children of the moon, which was born of the sun and earth. It is said that humans had great strength at the time and threatened to conquer the gods. The gods were then faced with the prospect of destroying the humans with lightning as they had done with the Titans but then they would lose the tributes given to the gods by humans. Zeus developed a creative solution by splitting humans in half as punishment for humanity's pride and doubling the number of humans who would give tribute to the gods. These split humans were in utter misery to the point where they would not eat and would perish so Apollo had sewn them up and reconstituted their bodies with the navel being the only remnant harkening back to their original form. Each human would then only have one set of genitalia and would forever long for his/her other half; the other half of his/her soul. It is said that when the two find each other, there is an unspoken understanding of one another, that they feel unified and would lay with each other in unity and would know no greater joy than that.
Twin Flames are still considered a real concept amongst some New Age followers. Some use the term Twin Flame merely to describe the unusually tight and overwhelming bond that is characteristic to some (sexual) lovers. Most strip the mythological origin of the words from their belief system, but still consider their Twin Flame to be a spouse chosen for / given to them by God.
Apart from Plato's Androgyne there is at least three other spiritual theories attempting to explain this love connection that to those describing it seems almost supernatural. These are Essence Twins of Michael Teachings  and (Kundalini) Mirror Spirits  The main difference to Twin Flames for both of these theories is that they do believe these bonds are potentially multiple.
The concept of Twin Rays should also be mentioned. There are several interpretations to what Twin Ray means, some believe it to be even stronger connection to Twin Flames that are a total of 12 grouped in two's (Twin Flame pairs) and then connected with a 13th person, who is a Twin Ray to all of them. Some believe a Twin Ray is the core essence of each Twin Flame pair, the part how they recognise each other.