We establish a general framework to explore parametric statistics of
individual energy levels in unitary random matrix ensembles. For a generic
confinement potential $W(H)$, we (i) find the joint distribution functions of
the eigenvalues of $H$ and $H'=H+V$ for an arbitrary fixed $V$ both for finite
matrix size $N$ and in the ``thermodynamic'' $N\to\infty$ limit; (ii) derive
many-point parametric correlation functions of the two sets of eigenvalues and
show that they are naturally parametrised by the eigenvalues of the reactance
matrix for scattering off the ``potential'' $V$; (iii) prove the universality
of the correlation functions in unitary ensembles with non-Gaussian
non-invariant confinement potential $W(H-V)$; (iv) establish a general scheme
for exact calculation of level-number-dependent parametric correlation
functions and apply the scheme to the calculation of intra-level velocity
autocorrelation function and the distribution of parametric level shifts.