Let (P2n*(z)) be a sequence of polynomials with real coefficients such that limnP2n*(eiϕ)=G(eiϕ) uniformly for ϕ∈[α-δ,β+δ] with G(eiϕ)≠0 on [α,β], where 0⩽α<β⩽π and δ>0. First it is shown that the zeros of pn(cosϕ)=Re{e-inϕP2n*(eiϕ)} are dense in [α,β], have spacing of precise order π/n and are interlacing with the zeros of pn+1(cosϕ) on [α,β] for every n⩾n0. Let (P˜2n*(z)) be another sequence of real polynomials with limnP˜2n*(eiϕ)=G˜(eiϕ) uniformly on [α-δ,β+δ] and G˜(eiϕ)≠0 on [α,β]. It is demonstrated that for all sufficiently large n the zeros of pn(cosϕ) and p˜n(cosϕ) strictly interlace on [α,β] if Im{G˜(eiϕ)/G(eiϕ)}≠0 on [α,β]. If the last expression is zero then a weaker kind of interlacing holds. These interlacing properties of the zeros are new for orthogonal polynomials also. For instance, for large n a simple criteria for interlacing of zeros of Jacobi polynomials on [-1+ɛ,1-ɛ], ɛ>0, is obtained. Finally it is shown that the results hold for wide classes of weighted Lq-minimal polynomials, q∈[1,∞], linear combinations and products of orthogonal polynomials, etc.