Does the flow of a vector field contain all and only its integral curves

Question

I have seen the flow of a vector field as the only local $\mathbb{R}$-action on a smooth manifold such that a handful of properties are verified (for example that $\forall p \in M$ $\theta_p:I\to M$ is a maximal integral curve or that $\frac{df(\theta (t=0,p))}{dt} = X_p (f)$). My question is: from visual intuition the flow basically describes the evolution of each point on the manifold subjected to the vector field, so intuitively I'd say that it's nothing less and nothing more than just the set of every integral curve. Is this correct or am I missing something