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This paper is concerned with stochastic differential equations (SDEs for short) with irregular coefficients. By utilising a functional analytic approximation approach, we establish the existence and uniqueness of strong solutions to a class of SDEs with critically irregular drift coefficients in a new critical Lebesgue space, where the element may be only weakly integrable in time. To be more precise, let b:[0,T]×R→R be Borel measurable, where T>0 is arbitrarily fixed and d⩾1. We consider the following SDE X=x+∫0tb(s,X)ds+W,t∈[0,T],x∈R, where {W} is a d-dimensional standard Wiener process. For p,q∈[1,+∞), we denote by C([0,T];L(R)) the space of all Borel measurable functions f such that [Formula presented]. If b=b+b such that |b(T−⋅)|∈C([0,T];L(R)) with 2/q+d/p=1 and ‖b(T−⋅)‖ is sufficiently small, and that b is bounded and Borel measurable, then we show that there exists a weak solution to the above equation, and if in addition [Formula presented], the pathwise uniqueness holds. Furthermore, we obtain the strong Feller property of the semi-group and the existence of density associated with the above SDE. Besides, we extend the classical results concerning partial differential equations (PDEs) of parabolic type with L(0,T;L(R)) coefficients to the case of parabolic PDEs with L(0,T;L(R)) coefficients, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Theorem 3.1). Our results extend Krylov-Röckner and Krylov's profound results of SDEs with singular time dependent drift coefficients [20,23] to the critical case of SDEs with critically irregular drift coefficients in a new critical Lebesgue space.