  We present the FastEMRIWaveforms (FEW) package, a collection of tools to
build and analyze extreme mass ratio inspiral (EMRI) waveforms. Here, we expand
on the Physical Review Letter that introduced the first fast and accurate
fully-relativistic EMRI waveform template model. We discuss the construction of
the overall framework; constituent modules; and the general methods used to
accelerate EMRI waveforms. Because the fully relativistic FEW model waveforms
are for now limited to eccentric orbits in the Schwarzschild spacetime, we also
introduce an improved Augmented Analytic Kludge (AAK) model that describes
generic Kerr inspirals. Both waveform models can be accelerated using graphics
processing unit (GPU) hardware. With the GPU-accelerated waveforms in hand, a
variety of studies are performed including an analysis of EMRI mode content,
template mismatch, and fully Bayesian Markov Chain Monte Carlo-based EMRI
parameter estimation. We find relativistic EMRI waveform templates can be
generated with fewer harmonic modes ($\sim10-100$) without biasing signal
extraction. However, we show for the first time that extraction of a
relativistic injection with semi-relativistic amplitudes can lead to strong
bias and anomalous structure in the posterior distribution for certain regions
of parameter space.
