  We study conformal field theories (CFTs) on curved spaces including both
orientable and unorientable manifolds possibly with boundaries. We first review
conformal transformations on curved manifolds. We then compute the identity
components of conformal groups acting on various metric spaces using a simple
fact; given local coordinate systems be single-valued. Boundary conditions thus
obtained which must be satisfied by conformal Killing vectors (CKVs) correctly
reproduce known conformal groups. As a byproduct, on $\mathbb
S^1_l\times\mathbb H^2_r$, by setting their radii $l=Nr$ with $N\in\mathbb
N^\times$, we find (the identity component of) the conformal group enhances,
whose persistence in higher dimensions is also argued. We also discuss forms of
correlation functions on these spaces using the symmetries. Finally, we study a
$d$-torus $\mathbb T^d$ in detail, and show the identity component of the
conformal group acting on the manifold in general is given by
$\text{Conf}_0(\mathbb T^d)\simeq U(1)^d$ when $d\ge2$. Using the fact, we
suggest some candidates of conformal manifolds of CFTs on $\mathbb T^d$ without
assuming the presence of supersymmetry (SUSY). In order to clarify which parts
of correlation functions are physical, we also discuss renormalization group
(RG) and local counterterms on curved spaces.
