Kali Yuga (Devanāgarī: कलियुग [kəli juɡə], lit. "age of Kali", or "age of vice") is the last of the four stages (or ages or yugas) the world goes through as part of a 'cycle of yugas' (i.e. Mahayuga) described in the Sanskrit scriptures. The other ages are called Satya Yuga, Treta Yuga, and Dvapara Yuga.
According to the Surya Siddhanta, Kali Yuga began at midnight (00:00) on 18 February 3102 BCE. This is also considered the date on which Lord Krishna left the earth to return to Vaikuntha. This information is placed at the temple of Bhalka, the place of this incident (see photo).
According to the astronomer and mathematician Aryabhatta the Kali Yuga started in 3102 BCE. He finished his book “Aryabhattiya” in 499 CE, in which he gives the exact year of the beginning of Kali Yuga. He writes that he wrote the book in the "year 3600 of the Kali Age" at the age of 23. As it was the 3600th year of the Kali Age when he was 23 years old, and given that Aryabhatta was born in 476 CE, the beginning of the Kali Yuga would come to (3600 - (476 + 23) + 1 (As only one year elapses between 1 BCE and 1 CE) = ) 3102 BCE.
According to KD Abhyankar, the starting point of Kaliyuga is an extremely rare planetary alignment, which is depicted in the Mohenjo-Daro seals. Going by this alignment the year 3102 BCE is slightly off. The actual date for this alignment is February 7 of 3104 BCE. There is also sufficient proof to believe that Vrdhha Garga knew of precession at least by 500 BCE. Garga had calculated the rate of precession to within 30% of what the modern scholars estimate.
Most interpreters of vedic scriptures, as Bhaktisiddhanta Sarasvati Goswami and his recent disciple Bhaktivedanta Swami Prabhupada believe that Earth is currently in Kali Yuga and lasts 432,000 years. Other authors, such as Swami Sri Yukteswar and Paramhansa Yogananda, believe that it is now an ascending Dvapara Yuga, indicating levels of cycles within each major Yuga period as each being a development, the smaller cycles within cycles eventually leading to full development of the qualities of the ages. The Kali Yuga is thought by some authors to last 6480 years although other durations have been proposed.
Hindus believe that human civilization degenerates spiritually during the Kali Yuga, which is referred to as the Dark Age because in it people are as far away as possible from God. Hinduism often symbolically represents morality (dharma) as an Indian bull. In Satya Yuga, the first stage of development, the bull has four legs, but in each age morality is reduced by one quarter. By the age of Kali, morality is reduced to only a quarter of that of the golden age, so that the bull of Dharma has only one leg.
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The Mahabharata War and the decimation of Kauravas thus happened at the "Yuga-Sandhi", the point of transition from one yuga to another. The scriptures mention Sage Narada to have momentarily intercepted the demon Kali on his way to the Earth when Duryodhana was about to be born in order to make him an embodiment of arishadvargas and adharma in preparation of the era of decay in values and the consequent havoc.
With regard to human relationships, Markandeya's discourse says:
The Brahma Vaivarta Purana (related to Rathantara kalpa) mentions a ten thousand-year period, starting from the traditional dating of the Kali Yuga epoch, during which bhakti yogis will be present. Lord Krishna foretold that Kali Yuga will be full of extreme hardships for people with ideals and values.
The Brahma-vaivarta Purana has words spoken by Lord Krishna to Mother Ganga just before the beginning of the Kali Yuga (the age of quarrel and strife). The Kali Yuga began approximately five thousand years ago, and it has a duration of 432,000 years, leaving us with 427,000 years until the end of the present age. Within this 432,000 year period, there is a period of 10,000 years that will be a golden age. That golden age is being described below by Lord Sri Krishna. Predicted in Brahma-vaivarta Purana 4.129. The fourth part of the Brahma-vaivarta is called Kṛṣṇa-janma-khanda. Chapter 129 is called Golokarohanam, because it describes how Krishna returns to His abode. This specific dialogue is between Lord Krishna and Mother Ganga. Verse 49 is a question by Ganga, verses 50–60 are Lord Sri Krishna's answer.
This text is taken from the Brahma-vaivarta Purana 
madbhaktāḥ santi bhūtale
madbhakteṣu gateṣu ca
"For 10,000 years of Kali such devotees of Mine will be present on earth. After the departure of My devotees there will be only one varna, Outcaste."
The above is supported in 4.90.32–33:
haris tiṣṭhati medinī
devānām pratimā pūjyā
śāstrāni ca purāṇakam
Hence to protect ourself from Kali, it is believed that we should start doing japa, mation or any yoga like Bhakti yoga, karma yoga, Raja yoga, and jnana yoga. But, Chanting the holy name of God is the best path in Kali yuga.
Kali is the reigning lord of Kali Yuga and his nemesis is Kalki, the tenth and final Avatar of Lord Vishnu. According to the Vishnu Purana, Kali is a negative manifestation working towards the cause of 'the end' or rather towards eventual rejuvenation of the universe. Kali also serves as an antagonistic force in the Kalki Purana. It is said that towards the end of this yuga, Kalki will return riding on a white horse to do battle with Kali and his dark forces. The world will suffer a fiery cataclysm that will destroy all evil, and a new age (the next Satya Yuga of the following Mahayuga), will begin.
Some Shaivites maintain that the ill effects of Kali Yuga can only be moderated by the manifestation of Lord Shiva Himself. Shastriji, one of the followers of Haidakhan Babaji, gave the following narration:
"Once Parvati asked Lord Shiva, her husband: 'You have done good work for the people in all ages, but I am afraid for the people in the Kali Yuga; how will they safeguard themselves?' Then Lord Shiva told Parvati: 'I will appear in the Kali Yuga and I will create a new state, a new centre of religion - a most important place, where I will live and establish all the Gods there.'"
Shastriji went further to suggest that this promise manifested through the person of Haidakhan Babaji. One of the central tenets of Haidakhan Babaji's teachings is the message of Karma Yoga or hard work. In the context of Kali yuga Haidakhan Babaji explained:
"As I have told you before, the thing needed in this Age is work (karma). In every Age people have reached salvation through different types of action and sadhana (spiritual discipline), but in this Age one can reach liberation only through hard work. I want real, practical human beings and only he is a true human being who lives in accordance with this Age. We need not consider religion or caste, but look only to hard work."
In Sikhism, Kali Yuga is metaphorically used to describe the state of the world as was commonly understood in the 16th century. It is stressed that one should mate as much as possible to reach the state of Nirvana and be liberated or be one with God. Guru Granth Sahib Ji on Ang:1185 says:
ab kaloo aaeiou rae : Now, the Dark Age of Kali Yuga has come.
eik naam bovahu bovahu : Plant the Naam, the Name of the One Lord.
an rooth naahee naahee : It is not the season to plant other seeds.
math bharam bhoolahu bhoolahu : Do not wander lost in doubt and delusion.
The Italian Traditionalist thinker Julius Evola, drawing on the work of René Guénon, describes modern Western civilization as being in its Kali Yuga phase, in a state of degeneration and eventual collapse, in his seminal 1934 anti-modernist work on the world of Tradition, Revolt Against the Modern World.
The dictionary definition of Kali Yuga at Wiktionary