We propose new structures called almost o-minimal structures and $\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open interval is a finite union of points and open intervals. The latter is a variant of van den Dries and Miller's analytic geometric categories and Shiota's $\mathfrak X$-sets and $\mathfrak Y$-sets. In them, the family of definable sets are closed only under proper projections unlike first-order structures. We demonstrate that an $\mathfrak X$-expansion of an ordered divisible abelian group always contains an o-minimal expansion of an ordered group such that all bounded $\mathfrak X$-definable sets are definable in the structure. Another contribution of this paper is a uniform local definable cell decomposition theorem for almost o-minimal expansions of ordered groups $\mathcal M=(M,

arXiv admin note: text overlap with arXiv:1912.05782