Here is a summary of time series in Julia. (Ported from this thread.)
This is a Julia version of CRAN Task View: Time Series Analysis .
TL;DR

ARCHModels is the most developed volatility model pkg in Julia
(it also fits ARMA models & does auto tuning)  StateSpaceModels is the most developed univariate TS pkg in Julia
 TSAnalysis is the most developed multivariate TS pkg in Julia
Please comment w/ relevant code I missed
1: Working with TS data in Julia:
Package  Description  Note 

TimeSeries  methods for working w TS data  
Dates  two types for working w dates: Date & DateTime

Standard Library 
Temporal 
TS class 
@dysonance 
Indicators  
RollingFunctions  unmaintained but useful 
2: Volatility models (univariate & multivariate):
Package  Description  Note 

ARCHModels  fit/sim ARMAGARCH models  @sbroda 
Financial Risk Forecasting  Textbook w code 
3: Univariate TS models:
Package  Description  Note 

StateSpaceModels  fit/sim SARIMA, filters & smoothers  @guilhermebodin 
AutoARIMA  Auto multiseasonal ARIMA  @pierrenodet 
ARFIMA  sim ARFIMA  @ Datseris 
RARIMA  wraps R  unmaintained 
ARMA  fit ARMA  @joefowler 
4: Multivariate TS models:
Package  Description  Note 

TSAnalysis  fit VARIMA, statespace  @fipelle 
Haroon Mumtaz  VAR w SV/TVP/SignRestrictions  often requested 
Dynamic Factor Models  FAVAR, SVAR  Stock & Watson 2016 
BusCycle Anatomy  VAR, VECM, IRF  Angeletos etal, aer 2020 
VectorAutoregressions  fit VAR, IRFs  @ lucabrugnolini 
VARmodels  fit VAR  unmaintained @tomaskrehlik 
VectorAR  VAR(p)  
FactorAugmentedVectorAR  FAVAR(p)  
Cointegration  Cointegration in VAR Models  @andreasnoack 
TVP_Julia  tvpVAR  
SignRestrictionVAR  signrestriction VAR 
5: TS Forecast Evaluation & Hypothesis Tests:
Package  Description  Note 

ForecastEval  dm/rc/mcs  @colintbowers 
HypothesisTests 
Uncategorized packages/links/etc for TS in Julia:
Package  Description  Note 

TSML  interface  @ppalmes 
ForecastingCombinations  @ lucabrugnolini  
TimeModels  fit ARIMA, GARCH  unmaintained 
ARMAProcesses  sim ARMA  
QuantEcon  
Creel Econometrics  Textbook  
Paul Soderlind  Teaching notes  
TimeSeriesClassification  MLJ interface  
SmoothLocalProjections  
TemporalGPs  Gaussian processes for TS  
ScoreDrivenModels  
DistributionalForecasts  
EntropyHub  
DCCA  
SerialDependence  
TimeseriesPrediction  
DependentBootstrap  
Forecast  STL decomposition  
X13  Wraps R  
SingularSpectrumAnalysis  
SMC  
StateSpaceRoutines  Filters, smoothers  
LowLevelParticleFilters  
Kalman  
HPFilter  
Hamilton Filter  
Dynamic Factor Models  Stock, Watson 2016  
CommonFactorModelStats  
FactorModels  used in thesis  unmaintained 
Factotum  Static factor models 