There are two main machine learning libraries in Julia, Flux.jl and Knet.jl
Flux.jl

using Flux;
using Flux.Tracker: update!;

// Create layers like this
inputSize = 10;
outputSize = 20;
myLayer = Dense(inputSize, outputSize, relu);
// You can get the weights using
myLayer.W;
myLayer.b;
// You can call the model to make a prediction
model(myData);
// Equivalent to 
relu(model.W * myData + model.b);


// Create Networks like this
model = Chain(
  Dense(10, 20, relu),
  Dense(20, 2)
);

// Calling the model will pass data through each layer.
model(myData);

// Get the parameters for the whole model with
p = params(model);

// Calculate the gradient with
gs = Tracker.gradient(function()
  predicted = model(myData);
  loss = sum((predicted - myLabels).^2)'
  return loss;
end, params(model));
// Gradient of layer 1 weights
gs[model[1].W];
// Update! will update the weights and clear the gradient
// Make sure to update all layers.
update!(model[1].W, -0.1 * gs[model[1[].W]);