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In this paper we show how to use simple partitioning lemmas in order to embedspanning graphs in a typical member of $G(n,p)$. Let the \emph{maximum density}of a graph $H$ be the maximum average degree of all the subgraphs of $H$.First, we show that for $p=\omega(\Delta^{12} n^{-1/2d}\log^3n)$, a graph$G\sim G(n,p)$ w.h.p.\ contains copies of all spanning graphs $H$ with maximumdegree at most $\Delta$ and maximum density at most $d$. For $d<\Delta/2$, thisimproves a result of Dellamonica, Kohayakawa, R\"odl and Ruci\'ncki. Next, weshow that if we additionally restrict the spanning graphs to have girth atleast 7 then the random graph contains w.h.p.\ all such graphs for$p=\omega(\Delta^{12} n^{-1/d}\log^3n)$. In particular, if$p=\omega(\Delta^{12} n^{-1/2}\log^3 n)$, the random graph therefore containsw.h.p.\ every spanning tree with maximum degree bounded by $\Delta$. Thisimproves a result of Johannsen, Krivelevich and Samotij.  Finally, in the same spirit, we show that for any spanning graph $H$ withconstant maximum degree, and for suitable $p$, if we randomly color the edgesof a graph $G\sim G(n,p)$ with $(1 + o(1))|E(H)|$ colors, then w.h.p.\ thereexists a \emph{rainbow} copy of $H$ in $G$ (that is, a copy of $H$ with alledges colored with distinct colors).

Universality of random graphs and rainbow embedding