The first part of this talk will be a gentle introduction to algebraic K-theory from a classical viewpoint. In the second part we will turn to the theme of ``motivic filtrations,'' in which K-theory and related invariants (such as cyclic homology) are decomposed into more manageable, geometric pieces. The talk will be complementary (but independent) to Antieau's colloquium of the previous week and will be aimed at non-specialists.