We study the \'etale sheafification of algebraic K-theory, called \'etale
K-theory. Our main results show that \'etale K-theory is very close to a
noncommutative invariant called Selmer K-theory, which is defined at the level
of categories. Consequently, we show that \'etale K-theory has surprisingly
well-behaved properties, integrally and without finiteness assumptions. A key
theoretical ingredient is the distinction, which we investigate in detail,
between sheaves and hypersheaves of spectra on \'etale sites.Comment: 89 pages, v3: various corrections and edit
