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Настоящая брошюра основана на лекциях и семинарах, которые автор вел на мехмате МГУ в 2004-2007 годах, в летней школе "Современная Математика" в 2007 году и на фивт МФТИ в 2013 году. В ней показано, как при решении интересных геометрических проблем, близких к приложениям, естественно возникают различные понятия кривизны, отличающей изучаемую геометрию от «обычной». Приведены прямые элементарные определения этих понятий. Брошюра предназначена студентам, аспирантам, работникам науки и образования, изучающим и применяющим дифференциальную геометрию. Для ее изучения достаточно владения основами анализа функций нескольких переменных (а во многих местах не нужно даже этого). Материал преподнесен в виде последовательностей задач. Это характерно не только для дзенских монастырей, но и для серьезного изучения математики. В третьем издании сделаны существенные улучшения.
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