We consider order and type properties of Marcinkiewicz and Lorentz function spaces. We show that if $0<p<1$, a $p$-normable quasi-Banach space is natural (i.e. embeds into a $q$-convex quasi-Banach lattice for some $q>0$) if and only if it is finitely representable in the space $L_{p,\infty}.$ We also show in particular that the weak Lorentz space $L_{1,\infty}$ do not have type $1$, while a non-normable Lorentz space $L_{1,p}$ has type $1$. We present also criteria for upper $r$-estimate and $r$-convexity of Marcinkiewicz spaces.