Let E be a countable-dimensional vector space over a countable field. We say that two infinite-dimensional subspaces of E are almost disjoint if their intersection is a finite-dimensional subspace. This allows us to define a notion of almost disjoint families in the context of vector spaces. We shall explore some combinatorial properties of mad families of vector spaces, along with their similarities and differences with mad families of subsets of natural numbers. If time permits, we discuss a few recent developments on the construction of full mad families of block subspaces.