File I/O
FITSIO.jl (wrapper for C library)
FITSFiles.jl (up & coming, all julia, but not as mature as FITSio.jl)
Data Wrangling
Math
Interpolations.jl (see also list of related packages near bottom of README)
Time series
Integration
QuadGK.jl: 1-d integration
Integrals.jl: unified interface for the numerical approximation of integrals
Statistics
Bayesian Inference
Machine Learning
Plotting
Coding practices
Astronomy-specific
Building blocks
Photometry
Imaging
Modeling
Octofitter.jl: Orbit fitting
Korg.jl: Stellar synthesis