How is IIT Madras ranked nationally and internationally?

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Different rankings say slightly different things, but the overall consensus is that IITM is right up there amongst the best. For instance, it is:

#1 in India since 2016, according to the National Institutional Ranking Framework (NIRF) of Higher Educational Establishments. In fact, the 2022 results just came out in mid-July and IITM continues to hold the top spot - in engineering and overall as well! This framework was approved by the MHRD and has been adopted by the Ministry of Education since 2015.
#1 in India since 2018 on the Atal Ranking of Institutions on Innovation Achievements, also known as the ARIIA rankings. This is an initiative of Ministry of Education to systematically rank all major higher educational institutions on indicators related to “Innovation and Entrepreneurship Development”.
#255 in the world according to the QS World University Rankings 2022, right after IITB, IITD, and IISc. These rankings are an annual publication by Quacquarelli Symonds which bases its results on the global overall ranking, the subject rankings, and five independent regional tables—namely Asia, Latin America, Emerging Europe and Central Asia, the Arab Region, and BRICS.
#17 in the world, as per the latest QS BRICS University Rankings, close behind IITB and IISc. This is an annual ranking of the top universities in the five BRICS countries – Brazil, Russia, India, China and South Africa. They use eight different indicators, including the ratio of faculty to students, the proportion of academic staff with a PhD and the institution’s reputation among academics and employers.
IITM is an Institute of Eminence (IoE). The IoE Scheme was launched by the Government of India to empower higher educational institutions and help them become world-class teaching and research institutions. IITM shares this proud distinction with only nine other institutes in the country and has set up multiple research initiatives to further its cause in pursuing world-class research.

Any other ranking you take will historically show IITM always at or close to the top!