Variational wave functions containing electronic pairing and suppressed
charge fluctuations (i.e., projected BCS states) have been proposed as the 
paradigm to understand strongly-correlated systems. I give a review of 
recent developments obtained in Hubbard-like models by considering long-range
Jastrow factor and backflow correlations or including magnetic order through
the Pfaffian ansatz.

In particular, I discuss how it is possible to have the metal-insulator
transition by using the Gutzwiller wave function supplemented with a long-range
Jastrow factor. An appealing interpretation in terms of the binding-unbinding
Kosterlitz-Thouless transition is obtained through the mapping onto a classical
model. I also show the role of backflow correlations in order to obtain a spin 
liquid insulator in the fermionic Hubbard model in presence of frustrating 
hopping amplitudes. The role of dimensionality is discussed on the bosonic 
Hubbard model, where the numerically exact solution can be obtained by 
Monte Carlo simulations. 

Projected wave functions turn out to be extremely accurate to describe
the low-energy properties of effective (frustrated) Heisenberg models.
Here, different quantum phases may be described with high accuracy by the same 
class of variational states, including dimerized and magnetically ordered 
states. Phases with magnetic order may be obtained from a straightforward 
generalization containing both antiferromagnetic and superconducting order 
parameters, as well as suitable spin Jastrow correlations. 

In summary, projected wave functions represent an highly flexible tool for 
understanding the physics of low-dimensional correlated materials.