  In the present paper, we study arrangements of smooth plane conics having
only nodes and tacnodes as the singularities. We provide an interesting
estimation on the number of nodes and tacnodes that depends only on a linear
function of the number of conics. Based on that result, we obtain a new upper
bound on the number of tacnodes which turns out to be better than Miyaoka's
bound for a large enough number of conics. We also study the freeness and
nearly freeness of such arrangements providing a detailed description.
