Let chi(n)(t) = (Sigma(n)(i=1) X-i(2)(t))(1/2), t >= 0 be a chi-process with n degrees of freedom where X (i) 's are independent copies of some generic centered Gaussian process X. This paper derives the exact asymptotic behaviour of
P{sup(t is an element of[0,T]) (chi(n)(t) - g(t) > u} as u -> infinity,
where T is a given positive constant, and g(a <...) is some non-negative bounded measurable function. The case g(t)equivalent to 0 has been investigated in numerous contributions by V.I. Piterbarg. Our novel asymptotic results, for both stationary and non-stationary X, are referred to as Piterbarg theorems for chi-processes with trend.