In this article we identify two large families of ideals of a Cohen-Macaulay (sometimes Gorenstein) local ring whose Rees algebras are Cohen-Macaulay. Our main results imply, for example, that if
(
R
,
M
)
(R,M)
is a regular local ring and
P
P
is a prime ideal of
R
R
such that
P
n
{P^n}
is unmixed for all
n
≥
1
n \geq 1
, then the Rees algebra
R
[
P
t
]
R[Pt]
is Cohen-Macaulay if either
dim
(
R
/
P
)
=
2
\dim (R/P) = 2
, or
dim
(
R
/
P
)
=
3
,
R
/
P
\dim (R/P) = 3,R/P
is Cohen-Macaulay, and
R
/
P
R/P
is integrally closed.