Siddhi (Sanskrit: सिद्धि siddhi; fulfillment, accomplishment) are spiritual, paranormal, supernatural, or otherwise magical powers, abilities, and attainments that are the products of spiritual advancement through sādhanās such as meditation and yoga. The term ṛddhi (Pali: iddhi, "psychic powers") is often used interchangeably in Buddhism.
The Visuddhimagga is one of the texts to give explicit details about how spiritual masters were thought to actually manifest supernormal abilities. It states that abilities such as flying through the air, walking through solid obstructions, diving into the ground, walking on water and so forth are achieved through changing one element, such as earth, into another element, such as air. The individual must master kasina meditation before this is possible. Dipa Ma, who trained via the Visuddhimagga, was said to demonstrate these abilities.
In Patañjali's Yoga Sutras IV.1 it is stated, Janma auṣadhi mantra tapaḥ samādhijāḥ siddhayaḥ, "Accomplishments may be attained through birth, the use of herbs, incantations, self-discipline or samadhi". Possible siddhis or siddhi-like abilities mentioned include:
In Shaivism, siddhis are defined as "Extraordinary powers of the soul, developed through consistent meditation and often uncomfortable and grueling tapas, or awakened naturally through spiritual maturity and yogic sādhanā." 
In the Bhagavata Purana, the five siddhis brought on by yoga and meditation are:
In the Samkhyakarika and Tattvasamasa, there are references to the attainment of eight siddhis by which "one becomes free of the pain of ignorance, one gains knowledge, and experiences bliss". The eight siddhis hinted at by Kapila in the Tattvasamasa are, as explained in verse 51 of the Samkhyakarika:
It is believed that the attainment of these eight siddhis renders one free of the pain of ignorance, and gives one knowledge and bliss.
In Sikhism, siddhi means "insight". "Eight Siddhis" is used for insight of the eight qualities of Nirankar or a.k.a. Akal Purakh mentioned in the Mul Mantar in the Guru Granth Sahib. God has eight qualities: EkOnkar, Satnam, Kartapurakh, Nirbhao, Nirvair, AkaalMurat, Ajooni and Svaibhang. The one who has insight of these qualities is called Sidh or Gurmukh.
In Tantric Buddhism, siddhi specifically refers to the acquisition of supernatural powers by psychic or magical means or the supposed faculty so acquired. These powers include items such as clairvoyance, levitation, bilocation, becoming as small as an atom, materialization, and having access to memories from past lives.