Summary: 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 \(b : [0, T] \times \mathbb{R}^d \to \mathbb{R}^d\) be Borel measurable, where \(T > 0\) is arbitrarily fixed and \(d \geqslant 1\). We consider the following SDE \[ X_t = x + \int\limits_0^t b(s, X_s) ds + W_t, \quad t \in [0, T], x \in \mathbb{R}^d, \] where \(\{W_t\}_{t \in [0, T]}\) is a \(d\)-dimensional standard Wiener process. For \(p, q \in [1, + \infty)\), we denote by \(\mathcal{C}_{[q]}([0, T]; L^p (\mathbb{R}^d))\) the space of all Borel measurable functions \(f\) such that \(t^{\frac{1}{q}} f(t) \in \mathcal{C}([0, T]; L^p(\mathbb{R}^d))\). If \(b = b_1 + b_2\) such that \(| b_1(T - \cdot) | \in \mathcal{C}_{[q]}([0, T]; L^p (\mathbb{R}^d))\) with \(2 / q + d / p = 1\) and \(\| b_1 (T - \cdot) \|_{\mathcal{C}_{[q]} ([0, T]; L^p (\mathbb{R}^d))}\) is sufficiently small, and that \(b_2\) is bounded and Borel measurable, then we show that there exists a weak solution to the above equation, and if in addition \(\lim_{t \downarrow 0} \| t^{\frac{1}{q}} b (T - t) \|_{L^p (\mathbb{R}^d)} = 0\), 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 \(L^q (0, T; L^p(\mathbb{R}^d))\) coefficients to the case of parabolic PDEs with \(L_{[q]}^\infty(0, T; L^p(\mathbb{R}^d))\) 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 [N. V. Krylov and M. Röckner, Probab. Theory Relat. Fields 131, No. 2, 154–196 (2005; Zbl 1072.60050); N. V. Krylov, Ann. Probab. 49, No. 6, 3142–3167 (2021; Zbl 1497.60076)] to the critical case of SDEs with critically irregular drift coefficients in a new critical Lebesgue space.