Chronology

Alan Baker is awarded a Fields Medal for his work on Diophantine equations.
Matiyasevich shows that "Hilbert's tenth problem" is unsolvable, namely that there is no general method for determining when polynomial equations have a solution in whole numbers.

Stephen Cook formulates the $P$ versus $NP$ problem regarding polynomial time algorithms.

Thom publishes Structural Stability and Morphogenesis which explains catastrophe theory. The theory examines situations in which gradually changing forces lead to so-called catastrophes, or abrupt changes, and has important applications in biology and optics.
Quillen formulates higher algebraic $K$ -theory, a new tool that uses geometric and topological methods and ideas to formulate and solve major problems in algebra, particularly ring theory and module theory.

Deligne proves the three "Weil conjectures".
Chen Jingrun shows that every sufficiently large even integer is the sum of a prime and a number with at most two prime factors. It makes a major contribution to the Goldbach Conjecture.

Feigenbaum discovers a new constant, approximately 4.669201609102..., which is related to period-doubling bifurcations and plays an important part in chaos theory.
Mandelbrot publishes Les objets fractals, forme, hasard et dimension which describes the theory of fractals.

Lakatos work Proofs and Refutations is published as a book two years after his death. First published in four parts in 1963-64 the work gives Lakatos's account of how mathematics develops.
Thurston is awarded the Oswald Veblen Geometry Prize of the American Mathematical Society for his work on foliations.
Appel and Haken show that the Four Colour Conjecture is true using 1200 hours of computer time to examine around 1500 configurations. (See this History Topic.)

Adleman, Rivest, and Shamir introduce public-key codes, a system for passing secret messages using large primes and a key which can be published.

Fefferman is awarded a Fields Medal for his work on partial differential equations, Fourier analysis, in particular convergence, multipliers, divergence, singular integrals and "Hardy spaces".
Mori proves the "Hartshorne conjecture", that projective spaces are the only smooth complete algebraic varieties with ample tangent bundles.