According to tradition, Hakka (Harihara) and Bukka (Bukkaraya), the first two rulers of the empire, were inspired by Vidyaranya. They established Vijayanagara with his blessings. Vidyaranya was the head of Sringeri Mutt.
There is a myth that Hakka and Bukka, while on a hunt, saw a hare fight a pack of hunting dogs. The brothers informed their guru Vidyaranya of this, who advised them to set up Vijayanagara (city of victory) in that spot.
Two theories abound on their origin. The first states that they were Kannada in origin, feudatories of the Hoysalas. They established Vijaynagara and power passed to them after the fall of Hoysalas. The other states that they were Telugu, in charge of the Kakatiya treasury. They fled after the defeat of Kakatiyas.

There are also other popular theories around their origin and one such story is that they were captured, converted and taken to Delhi and they escaped to reconvert and establish the empire.

The Wikimedia image is of an unknown painting that purportedly shows the brothers meeting Vidyaranya.
Source: https://pragyata.com/harihara-bukka-and-the-birth-of-vijayanagara/

In the 15^th century, there emerged a tradition of hundreds of ‘Haridāsas’ beginning with Vyāsarāya, Purandaradāsa and Kanakadāsa composing thousands of padas in Kannada. Purandaradāsa, is considered as Saṅgīta Pitāmahaḥ (Father of Carnatic Music) for having systematized the method of teaching music. His compositions rich in Śṛṅgāra, Adhyatma and Bhakti, packed the essence of all Vedas, Upaniṣads, Purāṇas in simple Kannada and his mudra ‘Purandara Viṭṭhala’ is a symbol of his devotion to the Viṭṭhala.

A maṇḍapa dedicated to Sri Purandara Dasa survives today in Hampi at the banks of Tungabhadra, besides the Vijaya Viṭṭhala Temple. The Wikimedia image by Ingo Menling is of the Sri Purandara Dasa Mantapa at Hampi.

Sayanacharya was a great Sanskrit scholar who lived during the reign of the first two Vijayanagara Kings. His commentaries and works have been the foundation for many translations of Vedas as of date. Every modern scholar of Vedas, Indian or Western, refers to his work obligatorily. There were many Vedic commentators before Sayana but all their work is lost. We know them by name only.
His work preserves and build upon these lost commentaries. While modern western scholars have deviated from his interpretation of Vedas, his work remains the standard Vedic commentary among traditional scholars. Some like Sri Aurobindo, have criticized him for placing greater emphasis on ritualism than spirituality in his work.
The image is of Max Muller’s translation of the Vedas, the first by a European. The title mentions that the book also has a translation of Sayana’s commentary.
Source: Dharmawiki.org, goldenreed.wordpress.com

Madura Vijayam or Virkamparaya Charitham is a Samskṛta Mahakavya written by the poetess Gangadevi. She was the wife of Kumara Kampanna, who was the son of emperor Bukka I. She chronicles her husband’s life in this work and describes the liberation of Madurai from the Madurai sultanate around 1370 CE.

She conjures vivid imagery of the heart wrenching conditions of the Srigangam and Madurai Temples and the Agrahāras under the Madurai Sultanate, “Before, Agrahāras were filled with smells of the smoke from sacrifices and the sounds of Vedic chanting, now, they are filled with smell of rotting meat and the drunken roars of the Turks“. Kampanna is then blessed by the Devi in his dream, and is handed over a divine sword of Lord Shiva, to go and vanquish the Turushkas.

Meenakshi Amman temple at Madurai is a living temple whose history can be traced back to the 4th century CE. The temple itself kept on being modified by later rulers. Within the temple complex is the magnificent Aiyaram Kal Mandapa (Thousand-Pillar Hall), which in fact contains 985 columns carved with divinities, female musicians, and attendant figures.
The image from Britannica com by Andrey Khrobostov shows the thousand pillared hall.
Source: https://www.britannica.com/topic/Meenaskshi-Amman-Temple

Translation credits: https://cbkwgl.wordpress.com/2018/07/07/madhura-vijayam-vijayanagar-conquest-of-madurai/

The most celebrated king of the Vijayanagara Empire, Krishna Devaraya, was a great patron of arts and literature. He was a polyglot himself, fluent in Kannada, Sanskrit, Telugu, and Tamil. His epic poem Āmuktamālyada (‘one who gave/offered a garland that was worn by her to The Lord.’), is about the wedding of Āṇḍāḷ and Mahāviṣṇu.

Āṇḍāḷ was one among the twelve Āḻvār, great Bhakti saints of the South and one of the most prominent woman bhaktas in history.  Godhāi (Āṇḍāḷ), raised by Periyāḻvār in deep bhakti, vowed to marry only Viṣṇu. Her devotion was fulfilled when Viṣṇu, as Raṅganātha of Śrīraṅgam, accepted her as a bride. She composed the Tiruppāvai and Nācciyār Tirumoḻi which are still sung and celebrated in the month of Mārgaḻi(Mārgaśīrṣa.

The image from Chiswich Auctions is of the Telugu manuscript of the book on English watermark paper from late 18th century CE.

The emblem of Vijayanagara, sometimes called Karnata Empire consisted of a boar (representing Varaha), the sun, the moon, and a dagger. The Chalukyan empire, also known as the Karnata Empire, which existed few hundred years prior, also had the Varaha /Boar as their emblem.
The empire’s emblem is not the only connection with Dashavataras of Vishnu. The kingdom of Kishkindha ruled by Bali and Sugreev and Hanuman’s birthplace is said to be modern Hampi.
The Wikimedia image is from Vijayanagara Sexcentenary Commemoration Volume by P N Bengeri published by ASI in 1936 showing a carving of the empire’s emblem.

Virūpākṣa (a form of Śiva) was the state deity of the Vijayanagara Empire. All their inscriptions and royal orders were signed with this name, symbolizing that the kings ruled on the deity’s behalf. Furthermore, the Virūpākṣa temple is the only living temple that survives in Hampi today. Here, Virūpākṣa is worshipped as the consort of Pampādevi, the personification of the Pampānadi and Pampātīrtha, the original name of Hampi in the Itihāsas and Purāṇas

The image from Epigraphia Indica Vol 17, p114 is of a royal Vijaynagara order.

The grand Vitthala temple is located in the heart of Hampi, near the banks of the Tungabhadra River. It is also known as Shree Vijay Vitthala Temple. It is a magnificent testament to the architectural prowess of the Vijayanagara Empire. It is a designated UNESCO World Heritage Site. It was constructed in the 15th century during the reign of King Krishnadevaraya II.
The temple has undergone architectural evolution with each successive king adding their unique touch to this captivating landmark. The temple is dedicated to Lord Vitthala, an incarnation of Lord Vishnu. It is renowned for its extraordinary stone structures, including the impressive Stone Chariot and the Musical Pillars.
The Wikimedia image is the stone chariot which figures in the 50 Rs note.
Source: https://www.incredibleindia.gov.in/en/karnataka/hampi/vitthala-temple-complex

Dussehra (celebrated as Mahanavami) was one of the biggest and most important festivals of the Vijayanagara Empire. It included a grand procession with decorated war elephants, cavalry, infantry, and tableaus. Travellers and traders like Niccolo de Conti, Abd-al-Razzaq, Fernao Nuniz and Domingo Paez have provided us with details about the festival.
Paez who was from Portugal was particularly mesmerized by the festival, “I have no words to express what I saw, with my head, so often turned from one side to another that I was almost following backwards off my horse with my senses lost.”
He was referring to the grand march of the army, which carried royal insignia and an emblem, held on the Vijayadashami day. The march was also an occasion to showcase the power and might of the dynasty.
The Wikimedia image by Kalyan Kumar shows crowds watching the elephant procession during Mysore Dussehra.
Source: https://www.deccanherald.com/india/karnataka/the-history-of-dasara-2712545

The Vijayanagara emperor Achyutadevaraya granted twelve villages to Kempegowda, one of his feudatory governors to establish a fort and a city. The governor went on to develop these villages into a flourishing city by constructing 65 pete-s, 50 lakes, several forts, and numerous temples. Today, after nearly 400 years, we know it as the Silicon City of India — Bengaluru.
The Wikimedia image by user C1MM is the Statue of Progress, 108-foot tall statue of Kempegowda near Kempegowda International Airport, Bengaluru.

In 1639, the East India Company obtained a small strip of land on the Coromandel Coast from the Vijayanagara king Pedda Venkata Raya. In 1645, the Company sent another mission to his successor and the last Vijayanagara king, Sriranga Raya, who issued a fresh grant with additional privileges. The land was obtained from Damarla Venkatappa Nayak, the local chieftain under the Vijayanagara Empire.
Within a year of the purchase, the Company laid the foundation of Fort St. George in 1640. This settlement would eventually grow into the thriving city of Madras, now Chennai. Incidentally, Madras came under the control of French briefly in 1746 during the global Anglo-French wars. It was returned to the British in 1748. The British would eventually win the war of colonialists in India, restricting the French to a few scattered outposts like Pondicherry.
The Wikimedia image is of a 1754 painting of Fort St. George by Jan Van Ryne.

At the Battle of Talikota in 1565, the Vijayanagara Empire faced a crushing defeat and the city of Vijayanagara was systematically looted and destroyed. It proved to be Vijayanagara’s last straw and it never recovered after its defeat.
Rama Raya, the power behind the throne in the empire, had defeated and humiliated the sultanates multiple times. They came together as a confederacy to avenge their defeats. Their armies formed a powerful coalition but even then, Vijayanagara forces were proving to be an equal match at Talikota.
Rama Raya had many mercenary brigades in his army and the battle turned decisively against him when the Gilani brothers, Nur Khan and Bijli Khan, switched sides in the middle of the battle.
The Wikimedia image is of a rare miniature from the Tazkiratul Mulk which depicts the beheading of Rama Raya after the battle.