Quantum Mechanics and Quantum Field Theory can both be formulated in terms of the so-called functional integrals.
The point is that intuitively it is an "integral over all possible paths" or rather "integral over all possible field configurations", traditionaly denoted as
$$\int A[\gamma(t)]\mathcal{D}\gamma(t)$$
$$\int A[\phi(x)]\mathcal{D}\phi(x)$$
for respectively paths and fields. It seems however that this is not well defined. I really don't understand how can one manipulate something that isn't defined, so I'm searching for the right way to understand these things.
Is there some way to make sense of these objects? I heard that as traditional measures it is not possible, but is there any other alternative way to make this be defined? If there is no way, how can someone work with one object that has no meaning associated with it and compute things with it?
Usually these are defined by a limiting process, splitting the QM path into lots of infinitesimal segments (or in Fourier space, by summing over a finite number of modes) and examining the continuous limit (if it exists). See for example the Feynman-Hibbs book. For field theory the path integral over field configurations is even more abstract and difficult to define.
If you don't like that, one may simply define the Gaussian functional integral in analogy to the discrete case (real variables and matrix in the exponent) and manipulate the path integral into moments of the Gaussian (essentially perturbation theory).
Incidentally, the worldline formalism to QFT replaces the functional field integral into a less abstract QM path integral.