We give a meaning to derivative of a function $uℝ\to X$, where $X$ is a complete metric space. This enables us to investigate differential equations in a metric space. One can prove in particular Gronwall’s Lemma, Peano and Picard Existence Theorems, Lyapunov Theorem or Nagumo Theorem in metric spaces. The main idea is to define the tangent space ${𝒯}_{x}X$ of $x\in X$. Let $u,v[0,1)\to X$, $u\left(0\right)=v\left(0\right)$ be continuous at zero. Then by the definition $u$ and $v$ are in the same equivalence class if they are tangent at zero, that is if $$\underset{h\to 0^{+}}{lim}\frac{d\left(u\right(h),v(h\left)\right)}{h}=0.$$ By ${𝒯}_{x}X$ we denote...