An almost \({\mathbf D}\)-structure (also named an almost para-complex structure) on a \(2n\)-dimensional manifold \(M\) is a \((1,1)\)-tensor field \(K\) such that \(K^2= \mathrm {Id}_{TM}\) and the two eigendistributions \(TM^+= \ker(\mathrm {Id}- K)\), \(TM^-= \ker(\mathrm {Id}+ K)\) have the same rank \(n\). In particular, if \(TM^+\), \(TM^-\) are involutive distributions, \(K\) is called a \({\mathbf D}\)-structure (para-complex structure). The aim of this paper is the development of the algebraic theory involved in the study of small deformations of \({\mathbf D}\)-structures on a compact \({\mathbf D}\)-manifold. Following [\textit{C. Medori} and \textit{A. Tomassini}, J. Noncommut. Geom. 5, No. 4, 507--522 (2011; Zbl 1235.53030)], given a compact \({\mathbf D}\)-manifold \((M,K)\), one introduces a differential graded lie algebra (DGLA) \(({\mathcal A},[[,]],\overline\partial_K)\). The deformations of \(K\) are parametrized by the 1-degree elements of \({\mathcal A}\) that satisfy the Maurer-Cartan equation. The author embeds \({\mathcal A}\) in another DGLA and restates the integrability condition of the deformations in terms of a new operator. In the second part of the paper, the analogous problem for CR \({\mathbf D}\)-structures on a compact contact manifold is considered. Finally, the author constructs families of CR \({\mathbf D}\)-structures and studies their deformations on two contact manifolds, namely the \(2n+1\)-dimensional Heisenberg group and a 5-dimensional compact quotient of a suitable nilpotent Lie group.