In the mathematical field of differential geometry, an almost-contact structure is a certain kind of geometric structure on a smooth manifold. Such structures were introduced by Shigeo Sasaki in 1960.

Precisely, given a smooth manifold , an almost-contact structure consists of a hyperplane distribution , an almost-complex structure on , and a vector field which is transverse to . That is, for each point of , one selects a codimension-one linear subspace of the tangent space , a linear map such that , and an element of which is not contained in .

Given such data, one can define, for each in , a linear map and a linear map by This defines a one-form and (1,1)-tensor field on , and one can check directly, by decomposing relative to the direct sum decomposition , that for any in . Conversely, one may define an almost-contact structure as a triple which satisfies the two conditions

• for any
• 

Then one can define to be the kernel of the linear map , and one can check that the restriction of to is valued in , thereby defining .

• David E. Blair. Riemannian geometry of contact and symplectic manifolds. Second edition. Progress in Mathematics, 203. Birkhäuser Boston, Ltd., Boston, MA, 2010. xvi+343 pp. ISBN 978-0-8176-4958-6, doi:10.1007/978-0-8176-4959-3
• Sasaki, Shigeo (1960). "On differentiable manifolds with certain structures which are closely related to almost contact structure, I". Tohoku Mathematical Journal. 12 (3): 459–476. doi:10.2748/tmj/1178244407.