  We develop techniques for defining and working with virtual fundamental
cycles on moduli spaces of pseudo-holomorphic curves which are not necessarily
cut out transversally. Such techniques have the potential for applications as
foundations for invariants in symplectic topology arising from "counting"
pseudo-holomorphic curves. We introduce the notion of an *implicit atlas* on a
moduli space, which is (roughly) a convenient system of local
finite-dimensional reductions. We present a general intrinsic strategy for
constructing a canonical implicit atlas on any moduli space of
pseudo-holomorphic curves. The main technical step in applying this strategy in
any particular setting is to prove appropriate gluing theorems. We require only
topological gluing theorems, that is, smoothness of the transition maps between
gluing charts need not be addressed. Our approach to virtual fundamental cycles
is algebraic rather than geometric (in particular, we do not use perturbation).
Sheaf-theoretic tools play an important role in setting up our functorial
algebraic "VFC package". We illustrate the methods we introduce by giving
definitions of Gromov--Witten invariants and Hamiltonian Floer homology over
$\mathbb Q$ for general symplectic manifolds. Our framework generalizes to the
$S^1$-equivariant setting, and we use $S^1$-localization to calculate
Hamiltonian Floer homology. The Arnold conjecture (as treated by Floer,
Hofer--Salamon, Ono, Liu--Tian, Ruan, and Fukaya--Ono) is a well-known
corollary of this calculation.
