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Our main results concern complete intersections of three real quadrics. Weprove that the maximal number $B^0_2(N)$ of connected components that a regularcomplete intersection of three real quadrics in $\Bbb{P}^N$ can have differs atmost by one from the maximal number of ovals of the submaximal depth$[(N-1)/2]$ of a real plane projective curve of degree $d=N+1$. As aconsequence, we obtain a lower bound \smash{$\frac14 N^2+O(N)$} and an upperbound \smash{$\frac38 N^2+O(N)$} for $B^0_2(N)$.

On the number of components of a complete intersection of real quadrics