  A Euclidean formulation of relativistic quantum mechanics for systems of a
finite number of degrees of freedom is discussed. Relativistic treatments of
quantum theory are needed to study hadronic systems at sub-hadronic distance
scales. While direct interaction approaches to relativistic quantum mechanics
have proved to be useful, they have two disadvantages. One is that cluster
properties are difficult to realize for systems of more than two particles. The
second is that the relation to quantum field theories is indirect. Euclidean
formulations of relativistic quantum mechanics provide an alternative
representation that does not have these difficulties. More surprising, the
theory can be formulated entirely in the Euclidean representation without the
need for analytic continuation. In this work a Euclidean representation of a
relativistic $N$-particle system is discussed. Kernels for systems of N free
particles of any spin are given and shown to be reflection positive. Explicit
formulas for generators of the Poincar\'e group for any spin are constructed
and shown to be self-adjoint on the Euclidean representation of the Hilbert
space. The structure of correlations that preserve both the Euclidean
covariance and reflection positivity is discussed.
