- Carnap publishes Logical Foundations of Probability.
- Hamming publishes a fundamental paper on error-detecting and error-correcting codes.
- Hodge puts forward the "Hodge Conjecture" on projective algebraic varieties.
- Serre uses spectral sequences to the study of the relations between the homology groups of fibre, total space and base space in a fibration. This enables him to discover fundamental connections between the homology groups and homotopy groups of a space and to prove important results on the homotopy groups of spheres.
- Serre is awarded a Fields Medal for his work on spectral sequences and his work developing complex variable theory in terms of sheaves.
- Kolmogorov publishes his second paper on the theory of dynamical systems. This marks the beginning of KAM-theory, which is named after Kolmogorov, Arnold and Moser.
- Cartan and Eilenberg develop homological algebra which allows powerful algebraic methods and topological methods to be related.
- Novikov proves the insolubility of the word problem for groups.
- Taniyama poses his conjecture on elliptic curves which will play a major role in the proof of Fermat's Last Theorem.
- Kolmogorov solves "Hilbert's Thirteenth Problem" on continuous functions of three variables which cannot be represented by continuous functions of two variables.
- Thom is awarded a Fields Medal for his work on topology, in particular on characteristic classes, cobordism theory and the "Thom transversality theorem".
- Boone proves that many decision problems for groups are insoluble.
- Marshall Hall publishes his famous text Theory of Groups.