The original seminal papers that linked K_0 to finiteness of a CW complex and K_1 to simple homotopy theory:

• Finiteness conditions for CW-complexes
  Wall, 1965.

• Finiteness conditions for CW-complexes II
  Wall, 1966.
  The first paper defines the obstruction in K_0 for finiteness of a CW complex. The second paper redoes the first in terms chain complexes and defines the obstruction as an Euler characteristic, giving the modern interpretation of the Wall finiteness obstruction.

• Whitehead torsion
  Milnor, 1966.
  This is a Bulletin article, i.e., a long expository paper of the results in several papers of Whitehead on simple homotopy types and Whitehead torsion, and the independent results of Smale, Barden and Stallings generalizing the h-cobordism theorem of Smale to the non-simply connected case. This is the beginning of the relation of K-theory to geometric topology (it’s just about K_1).

• Topological invariance of Whitehead torsion
  Chapman, 1974.
  In this paper Chapman proves topological invariance of Whitehead torsion, proving the conjecture posed in Whitehead’s paper.

Expository accounts of this material from a more modern point of view:

• A Course in Simple-Homotopy Theory
  M. M. Cohen, 1970.

• A survey of Wall's finiteness obstruction
  Steve Ferry and Andrew Ranicki, 2000.

• The Baum–Connes and the Farrell–Jones Conjectures in K- and L-Theory
  Wolfgang Luck and Holger Reich, 2004.

• Lecture Notes on Algebraic K-theory (Section 1.11)
  John Rognes, 2010.

• Understanding the Wall finiteness obstruction (blog post)
  Akhil Mathew, 2012.

• Connections of K-Theory to Geometry and Topology
  Maxim Stykow, 2013.

• The Wall finiteness obstruction, Whitehead torsion, and Whitehead torsion II
  from Jacob Lurie's class on K-theory and manifolds, 2014.

• K-theory and Geometric Topology
  Jonathan Rosenberg - chapter for the Handbook of Homotopy Theory, 2019.

• Simple homotopy theory and Whitehead torsion
  Richard Wong.

• Notes on algebraic torsion
  Stefan Friedl.

The main constructions of higher K-groups use the +-construction, the Q-construction, or Waldhausen categories. See also Chapter IV of Weibel's K-book.

Q-construction and the +-construction

• Higher Algebraic K-theory, I
  Daniel Quillen, 1973.

• Higher Algebriac K-theory, II [After Daniel Quillen]
  Daniel Grayson, 1976.

• On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field
  Daniel Quillen, 1972.
  This is a (very advanced) paper. This is the original definition of the +-construction, and has as an application the computation of the K-theory of finite fields.

Waldhausen categories

• Algebraic K-theory of spaces
  Fridehelm Waldhausen, 1985.
  This is the original source material.

• Lecture Notes on Algebraic K-theory
  John Rognes, 2010.

• K-theory of a Waldhausen Category as a symmetric spectrum
  Mitya Boyarchenko, 2006.

• On fundamental theorems in algebraic K-theory
  Randy, McCarthy, 1993.
  A more basic proof of the (foundational and extremely important) proof of additivity.

See also Chapter V of Weibel's K-book.

Milnor K-theory and the Milnor Conjecture

• Algebraic K-theory
  John Milnor, 1972.

• Algebraic K-theory and quadratic forms
  John Milnor, 1970.
  Milnor's Conjecture relating Milnor K-theory and étale cohomology first appeared here.

• Motivic cohomology with Z/2 coefficients
  Vladamir Voevodsky, 2001.
  Proves the Z/2 coefficients version of Milnor's conjecture relating Milnor K-theory and étale cohomology.

More resources on the Milnor conjecture:

• Developments in algebraic K-theory and quadratic forms after the work of Milnor
  Alexander Merkurjev, 2010.

• Voevodsky's proof of the Milnor conjecture
  Fabien Morel, 1998

• Notes on the Milnor conjectures
  Dan Dugger, 2004.

Descent in algebraic K-theory

The foundational papers on the relationship of algebraic K-theory and algebraic geometry:

• Algebraic K-theory and etale cohomology
  R. W. Thomason, 1980.

• Higher Algebraic K-theory of schemes and derived categories
  R. W. Thomason and Thomas Trobaugh, 1990.

Some further resources on topics of interest:

• Hypercohomology spectra and Thomason's descent theorem
  Stephen Mitchell, 1997.

• On the Lichtenbaum--Quillen Conjectures from a Stable Homotopy-Theoretic Viewpoint
  Stephen Mitchell, 1994.

• Galois Cohomology and Algebraic K-theory of finite fields
  Gabe Angelini-Knoll, 2013.

Vandiver conjecture

Unfortunately, there is no main paper on this conjecture because no one knows how to solve it yet.

• Kurihara: Some remarks on conjectures about cyclotomic fields and K-groups of Z
  One of the first papers explaining the equivalence between the Vandiver conjecture and algebraic K-theory of the integers.

• http://www.math.tifr.res.in/~eghate/vandiver.pdf Notes from a summer school about the Vandiver conjecture and K-theory formulation:

• Gajda: n K_*(Z) and classical conjectures in the arithmetic of cyclotomic fields . A survey.

• Kolster: K-theory and Arithmetic

• Weibel: Algebraic K-theory of rings and integers in local and global fields. A survey of what is known about the algebraic K-theory of fields.

The Quillen-Lichtenbaum Conjecture

• Lichtenbaum, Values of zeta functions, etale cohomology and algebraic K-theory

• Clark Barwick’s notes: https://www.maths.ed.ac.uk/~cbarwick/papers/descentK.pdf

• The article by Kolster linked above also address this.

• Cyclic Homology
  Jean-Louis Loday, 1998.
  The canonical reference for the homological side of trace methods, Hochschild homology (HH) and its variants cyclic homology (HC) and negative cyclic homology (HC^-).

• Algebraic K-theory and traces
  Ib Madsen, 1995.
  This survey contains many of the classical constructions and applications of topological Hochschild homology (THH), topological cyclic homology (TC), and the cyclotomic trace map from algebraic K-theory to TC. It is more accessible than the classic papers by Bökstedt and by Bökstedt, Hsiang, and Madsen, but covers the same material.

• The local structure of algebraic K-theory
  Bjorn Dundas, Tom Goodwillie, and Randy McCarthy, 2012.
  This book contains a great amount of foundational material for THH, TC, the cyclotomic trace, and the Dundas-McCarthy Theorem.

• On topological cyclic homology
  Thomas Nikolaus and Peter Scholze, 2017.
  This paper recast the classical definition of TC in terms of the Tate construction, avoiding the use of genuine G-spectra entirely. It has led to significant simplifications in the calculation of TC in recent years.

• Topological cyclic homology
  Lars Hesselholt and Thomas Nikolaus, 2019.
  A survey paper covering recent developments in the field.

• Waldhausen, An overview of how manifolds relate to algebraic K-theory
  An expository paper by Waldhausen overviewing the stable parametrized h-cobordism theorem.

• Waldhausen, Jahren, Rognes, Spaces of PL-manifolds and categories of simple maps
  A complete overview and proof of the theorem. This is long and technical, but describes the precise connections between K-theory and cobordism.

• Connections of K-Theory to Geometry and Topology
  Section 4.5 is of particular interest here.

• K-theory and Geometric Topology
  Jonathan Rosenberg - chapter for the Handbook of Homotopy Theory, 2019.

• On the universal property of Waldhausen’s K-theory
  Wolfgang Steimle, 2017.
  This paper gives a direct and short (8 pages!) characterization of the S.-construction as the universal functor that satisfies additivity.

• Universal characterization of higher algebraic K-theory
  Andrew Blumberg, David Gepner, and Gonçalo Tabuada, 2013.
  This gives a characterization of algebraic K-theory as the universal additive functor in the setting of small stable infinity categories.

• On the algebraic K-theory of higher categories
  Clark Barwick, 2012.
  This describes Waldhausen K-theory as a Goodwillie differential and also gives a universal characterization of algebraic K-theory.

• On the K-theory of pullbacks
  Land and Tamme, 2018.
  This paper completely solves the excision problem in K-theory. An excision sequence for a pullback of rings was known not to extend past K_2, but the authors show that if you work with ring spectra instead of rings, you can define such a sequence.