For m=3,4,..., the polygonal numbers of order m are given by
$p_m(n)=(m-2)n(n-1)/2+n (n=0,1,2,...)$. For positive integers $a,b,c$ and
$i,j,k>2$ with max{i,j,k}>4, we call the triple $(ap_i,bp_j,cp_k)$ universal if
for any n=0,1,2,... there are nonnegative integers $x,y,z$ such that
$n=ap_i(x)+bp_j(y)+cp_k(z)$. We show that there are only 95 candidates for
universal triples (two of which are $(p_4,p_5,p_6)$ and $(p_3,p_4,p_{27})$),
and conjecture that they are indeed universal triples. For many triples
$(ap_i,bp_j,cp_k)$ (including (p_3,4p_4,p_5), (p_4,p_5,p_6) and (p_4,p_4,p_5)),
we prove that any nonnegative integer can be written in the form
$ap_i(x)+bp_j(y)+cp_k(z)$ with $x,y,z\in\Z$. We also show some related new
results on ternary quadratic forms, one of which states that any nonnegative
integer n=1(mod 6) can be written in the form $x^2+3y^2+24z^2$ with
$x,y,z\in\Z$. In addition, we pose several related conjectures one of which
states that for any m=3,4,... each natural number can be written as
p_{m+1}(x_1)+p_{m+2}(x_2)+p_{m+3}(x_3)+r with x_1,x_2,x_3 nonnegative integers
and r among 0,...,m-3.