In this talk, we will study algebraic K-groups of varieties over finite fields, local fields, and more generally higher local fields. These groups were first defined during the 1950s and 1960s by A. Grothendieck, J. Whitehead, H. Bass, and J. Milnor. Later, in the 1970s, D. Quillen defined higher algebraic K-groups Kn(X), for rings and, more generally, schemes, which encode arithmetic and geometric information about the underlying scheme.
We will begin by recalling the definitions and basic properties of these groups and reviewing classical results. In the latter part of the talk, I will present new results on these groups for varieties over finite fields, local fields, and more generally higher local fields, which I obtained in joint work with R. Gupta and A. Krishna.
