In the first part of this talk I will introduce Haagerup's theory of noncommutative L_p-spaces using the nice algebraic formalism of modular algebras by Yamagami. (Here L_p=L^{1/p}, in particular, L_0=L^\infty and L_1/2=L2.) Then I will discuss some interesting properties of the resulting L_p-spaces, in particular I will prove the following theorem: L_p(M)\otimes_M L_q(M)=L_p+q(M) for an arbitrary von Neumann algebra~M and arbitrary complex p and q with nonnegative real parts. Equality here means isometric isomorphism of M-M-bimodules. In the second part of the talk I will describe L_p-modules by Junge and Sherman, which are the noncommutative analogs of modules of p-sections of bundles of Hilbert spaces over a measurable space. The special cases p=0 and p=1/2 correspond to the well-known cases of Hilbert W*-modules and Connes' correspondences. I will prove that W*-categories of L_p-modules for all values of~p are equivalent to each other. After that I will explain how Connes' fusion (and its generalized version), which originally had very technical definition, can be described easily in this algebraic formalism.