Let $$n\in \mathbb {N}$$ n ∈ N , $$n\ge 2$$ n ≥ 2 , $$q\in (1,\infty ]$$ q ∈ ( 1 , ∞ ] and let $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n be an open bounded set. We obtain sharp constants concerning the Moser-type inequalities corresponding to the Lorentz-Sobolev space $$W_0^1L^{n,q}(\Omega )$$ W 0 1 L n , q ( Ω ) equipped with the norm $$\begin{aligned} \Vert \nabla u\Vert _{(n,q)}:= {\left\{ \begin{array}{ll}\Vert t^{\frac{1}{n}-\frac{1}{q}}|\nabla u|^{**}(t)\Vert _{L^q((0,\infty ))}&{}\text {for }q\in (1,\infty )\\ \sup _{t\in (0,\infty )}t^{\frac{1}{n}}|\nabla u|^{**}(t)&{}\text {for }q=\infty . \end{array}\right. } \end{aligned}$$ ‖ ∇ u ‖ ( n , q ) : = ‖ t 1 n - 1 q | ∇ u | ∗ ∗ ( t ) ‖ L q ( ( 0 , ∞ ) ) for q ∈ ( 1 , ∞ ) sup t ∈ ( 0 , ∞ ) t 1 n | ∇ u | ∗ ∗ ( t ) for q = ∞ . We also derive the key estimate for the Concentration-Compactness Principle in the case $$q\in (1,\infty )$$ q ∈ ( 1 , ∞ ) with respect to the above norm.