In this paper we study the transport equation in $\mathbb{R}^{n} \times (0,T)$, $T >0$, $n\ge 2$, $$ \partial \_t f + v\cdot \nabla f = g, \quad f(\cdot,0)= f\_0 \quad \text{in }\mathbb{R}^{n}, $$ in generalized Campanato spaces $\mathscr{L}^{s}{{q(p, N)}}(\mathbb{R}^{n})$. The critical case is particularly interesting, and is applied to the local well-posedness problem for the incompressible Euler equations in a space close to the Lipschitz space in our companion paper \[Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 2, 201–241]. In the critical case $s=q=N=1$, we have the embeddings $B^{1}{\infty, 1} (\mathbb R^n) \hookrightarrow \mathscr{L}^{1}{1(p, 1)}(\mathbb {R}^{n}) \hookrightarrow C^{0, 1} (\mathbb R^n)$, where $B^{1}{\infty, 1} (\mathbb R^n)$ and $C^{0, 1} (\mathbb R^n)$ are the Besov and Lipschitz spaces, respectively. For $f\_0\in \mathscr{L}^{1}{1(p, 1)}(\mathbb {R}^{n})$, $v\in L^1(0,T; \mathscr {L}^{1}{1(p, 1)}(\mathbb {R}^{n})))$ and $g\in L^1(0,T; \mathscr {L}^{1}{1(p, 1)}(\mathbb {R}^{n})))$, we prove the existence and uniqueness of solutions to the transport equation in $L^\infty(0,T; \smash{\mathscr{L}^{1}{1(p, 1)}(\mathbb {R}^{n})})$ such that $$ |f|{L^\infty(0,T; \mathscr{L}^{1}{1(p, 1)}(\mathbb {R}^{n})))} \le C \big( |v|{L^1(0,T; \mathscr{L}^{1}{1(p, 1)}(\mathbb {R}^{n})))}, |g|{L^1(0,T; \mathscr {L}^{1}{1(p, 1)} (\mathbb {R}^{n})))}\big). $$ Similar results for the other cases are also proved.