diff --git a/docs/src/apireference.md b/docs/src/apireference.md
index 48e01eec4b..acbec37714 100644
--- a/docs/src/apireference.md
+++ b/docs/src/apireference.md
@@ -74,6 +74,15 @@ AbstractSubmittable
 submit
 ```
 
+List of submittables
+
+```@docs
+LazyConstraint
+HeuristicSolutionStatus
+HeuristicSolution
+UserCut
+```
+
 ## Model Interface
 
 ```@docs
@@ -121,6 +130,10 @@ SolverName
 Silent
 TimeLimitSec
 RawParameter
+AbstractCallback
+LazyConstraintCallback
+HeuristicCallback
+UserCutCallback
 ```
 
 List of attributes useful for optimizers
@@ -219,6 +232,7 @@ Calls to `get` and `set` should include as an argument a single `VariableIndex`
 VariableName
 VariablePrimalStart
 VariablePrimal
+CallbackVariablePrimal
 ```
 
 ### Constraints
@@ -438,6 +452,12 @@ LowerBoundAlreadySet
 UpperBoundAlreadySet
 ```
 
+As discussed in [`AbstractCallback`](@ref), trying to [`get`](@ref) attributes
+inside a callback may throw:
+```@docs
+OptimizeInProgress
+```
+
 The rest of the errors defined in MOI fall in two categories represented by the
 following two abstract types:
 ```@docs
diff --git a/src/attributes.jl b/src/attributes.jl
index 930a894280..cc013c6ed2 100644
--- a/src/attributes.jl
+++ b/src/attributes.jl
@@ -377,6 +377,14 @@ function throw_set_error_fallback(model::ModelLike,
     end
 end
 
+"""
+    SettingSingleVariableFunctionNotAllowed()
+
+Error type that should be thrown when the user calls [`set`](@ref) to change
+the [`ConstraintFunction`](@ref) of a [`SingleVariable`](@ref) constraint.
+"""
+struct SettingSingleVariableFunctionNotAllowed <: Exception end
+
 """
     submit(optimizer::AbstractOptimizer, sub::AbstractSubmittable,
            values...)::Nothing
@@ -397,13 +405,92 @@ function submit(model::ModelLike, sub::AbstractSubmittable, args...)
     end
 end
 
+## Submittables
+
 """
-    SettingSingleVariableFunctionNotAllowed()
+    LazyConstraint(callback_data)
+
+Lazy constraint `func`-in-`set` submitted as `func, set`. The optimal
+solution returned by [`VariablePrimal`](@ref) will satisfy all lazy
+constraints that have been submitted.
+
+This can be submitted only from the [`LazyConstraintCallback`](@ref). The
+field `callback_data` is a solver-specific callback type that is passed as the
+argument to the feasible solution callback.
+
+## Examples
 
-Error type that should be thrown when the user [`set`](@ref) the
-[`ConstraintFunction`](@ref) of a [`SingleVariable`](@ref) constraint.
+Suppose `fx = MOI.SingleVariable(x)` and `fx = MOI.SingleVariable(y)`
+where `x` and `y` are [`VariableIndex`](@ref)s of `optimizer`. To add a
+`LazyConstraint` for `2x + 3y <= 1`, write
+```julia
+func = 2.0fx + 3.0fy
+set = MOI.LessThan(1.0)
+MOI.submit(optimizer, MOI.LazyConstraint(callback_data), func, set)
+```
+inside a [`LazyConstraintCallback`](@ref) of data `callback_data`.
 """
-struct SettingSingleVariableFunctionNotAllowed <: Exception end
+struct LazyConstraint{CallbackDataType} <: AbstractSubmittable
+    callback_data::CallbackDataType
+end
+
+"""
+    HeuristicSolutionStatus
+
+An Enum of possible return values for [`submit`](@ref) with
+[`HeuristicSolution`](@ref).
+This informs whether the heuristic solution was accepted or rejected.
+Possible values are:
+* `HEURISTIC_SOLUTION_ACCEPTED`: The heuristic solution was accepted.
+* `HEURISTIC_SOLUTION_REJECTED`: The heuristic solution was rejected.
+* `HEURISTIC_SOLUTION_UNKNOWN`: No information available on the acceptance.
+"""
+@enum(HeuristicSolutionStatus,
+      HEURISTIC_SOLUTION_ACCEPTED,
+      HEURISTIC_SOLUTION_REJECTED,
+      HEURISTIC_SOLUTION_UNKNOWN)
+
+"""
+    HeuristicSolution(callback_data)
+
+Heuristically obtained feasible solution. The solution is submitted as
+`variables, values` where `values[i]` gives the value of `variables[i]`,
+similarly to [`set`](@ref). The [`submit`](@ref) call returns a
+[`HeuristicSolutionStatus`](@ref) indicating whether the provided solution
+was accepted or rejected.
+
+This can be submitted only from the [`HeuristicCallback`](@ref). The
+field `callback_data` is a solver-specific callback type that is passed as the
+argument to the heuristic callback.
+
+Some solvers require a complete solution, others only partial solutions.
+"""
+struct HeuristicSolution{CallbackDataType} <: AbstractSubmittable
+    callback_data::CallbackDataType
+end
+
+"""
+    UserCut(callback_data)
+
+Constraint `func`-to-`set` suggested to help the solver detect the solution
+given by [`CallbackVariablePrimal`](@ref) as infeasible. The cut is submitted
+as `func, set`.
+Typically [`CallbackVariablePrimal`](@ref) will violate integrality constraints,
+and a cut would be of the form [`ScalarAffineFunction`](@ref)-in-[`LessThan`](@ref)
+or [`ScalarAffineFunction`](@ref)-in-[`GreaterThan`](@ref). Note that, as
+opposed to [`LazyConstraint`](@ref), the provided constraint cannot modify the
+feasible set, the constraint should be redundant, e.g., it may be a consequence
+of affine and integrality constraints.
+
+This can be submitted only from the [`UserCutCallback`](@ref). The
+field `callback_data` is a solver-specific callback type that is passed as the
+argument to the infeasible solution callback.
+
+Note that the solver may silently ignore the provided constraint.
+"""
+struct UserCut{CallbackDataType} <: AbstractSubmittable
+    callback_data::CallbackDataType
+end
 
 ## Optimizer attributes
 
@@ -461,6 +548,137 @@ struct RawParameter <: AbstractOptimizerAttribute
     name::Any
 end
 
+### Callbacks
+
+"""
+    struct OptimizeInProgress{AttrType<:AnyAttribute} <: Exception
+        attr::AttrType
+    end
+
+Error thrown from optimizer when `MOI.get(optimizer, attr)` is called inside an
+[`AbstractCallback`](@ref) while it is only defined once [`optimize!`](@ref) has
+completed. This can only happen when `is_set_by_optimize(attr)` is `true`.
+"""
+struct OptimizeInProgress{AttrType<:AnyAttribute} <: Exception
+    attr::AttrType
+end
+
+function Base.showerror(io::IO, err::OptimizeInProgress)
+    print(io, typeof(err), ": Cannot get result as the `MOI.optimize!` has not",
+          " finished.")
+end
+
+"""
+    abstract type AbstractCallback <: AbstractOptimizerAttribute end
+
+Abstract type for optimizer attribute representing a callback function. The
+value set to subtypes of `AbstractCallback` is a function that may be called
+during [`optimize!`](@ref). As [`optimize!`](@ref) is in progress, the result
+attributes (i.e, the attributes `attr` such that `is_set_by_optimize(attr)`)
+may not be accessible from the callback, hence trying to get result attributes
+might throw a [`OptimizeInProgress`](@ref) error.
+
+At most one callback of each type can be registered. If an optimizer already
+has a function for a callback type, and the user registers a new function,
+then the old one is replaced.
+
+The value of the attribute should be a function taking only one argument,
+commonly called `callback_data`, that can be used for instance in
+[`LazyConstraintCallback`](@ref), [`HeuristicCallback`](@ref) and
+[`UserCutCallback`](@ref).
+"""
+abstract type AbstractCallback <: AbstractOptimizerAttribute end
+
+"""
+    LazyConstraintCallback() <: AbstractCallback
+
+The callback can be used to reduce the feasible set given the current primal
+solution by submitting a [`LazyConstraint`](@ref). For instance, it may be
+called at an incumbent of a mixed-integer problem. Note that there is no
+guarantee that the callback is called at *every* feasible primal solution.
+
+The feasible primal solution is accessed through
+[`CallbackVariablePrimal`](@ref). Trying to access other result
+attributes will throw [`OptimizeInProgress`](@ref) as discussed in
+[`AbstractCallback`](@ref).
+
+## Examples
+
+```julia
+x = MOI.add_variables(optimizer, 8)
+MOI.set(optimizer, MOI.LazyConstraintCallback(), callback_data -> begin
+    sol = MOI.get(optimizer, MOI.CallbackVariablePrimal(callback_data), x)
+    if # should add a lazy constraint
+        func = # computes function
+        set = # computes set
+        MOI.submit(optimizer, MOI.LazyConstraint(callback_data), func, set)
+    end
+end
+```
+"""
+struct LazyConstraintCallback <: AbstractCallback end
+
+"""
+    HeuristicCallback() <: AbstractCallback
+
+The callback can be used to submit [`HeuristicSolution`](@ref) given the
+current primal solution.
+For instance, it may be called at fractional (i.e., non-integer) nodes in the
+branch and bound tree of a mixed-integer problem. Note that there is not
+guarantee that the callback is called *everytime* the solver has an infeasible
+solution.
+
+The current primal solution is accessed through
+[`CallbackVariablePrimal`](@ref). Trying to access other result
+attributes will throw [`OptimizeInProgress`](@ref) as discussed in
+[`AbstractCallback`](@ref).
+
+## Examples
+
+```julia
+x = MOI.add_variables(optimizer, 8)
+MOI.set(optimizer, MOI.HeuristicCallback(), callback_data -> begin
+    sol = MOI.get(optimizer, MOI.CallbackVariablePrimal(callback_data), x)
+    if # can find a heuristic solution
+        values = # computes heuristic solution
+        MOI.submit(optimizer, MOI.HeuristicSolution(callback_data), x,
+                   values)
+    end
+end
+```
+"""
+struct HeuristicCallback <: AbstractCallback end
+
+"""
+    UserCutCallback() <: AbstractCallback
+
+The callback can be used to submit [`UserCut`](@ref) given the current primal
+solution. For instance, it may be called at fractional (i.e., non-integer) nodes
+in the branch and bound tree of a mixed-integer problem. Note that there is not
+guarantee that the callback is called *everytime* the solver has an infeasible
+solution.
+
+The infeasible solution is accessed through
+[`CallbackVariablePrimal`](@ref). Trying to access other result
+attributes will throw [`OptimizeInProgress`](@ref) as discussed in
+[`AbstractCallback`](@ref).
+
+## Examples
+
+```julia
+x = MOI.add_variables(optimizer, 8)
+MOI.set(optimizer, MOI.UserCutCallback(), callback_data -> begin
+    sol = MOI.get(optimizer, MOI.CallbackVariablePrimal(callback_data), x)
+    if # can find a user cut
+        func = # computes function
+        set = # computes set
+        MOI.submit(optimizer, MOI.UserCut(callback_data), func, set)
+    end
+end
+```
+"""
+struct UserCutCallback <: AbstractCallback end
+
 ## Model attributes
 
 """
@@ -681,6 +899,16 @@ struct VariablePrimal <: AbstractVariableAttribute
 end
 VariablePrimal() = VariablePrimal(1)
 
+"""
+    CallbackVariablePrimal(callback_data)
+
+A variable attribute for the assignment to some primal variable's value during
+the callback identified by `callback_data`.
+"""
+struct CallbackVariablePrimal{CallbackDataType} <: AbstractVariableAttribute
+    callback_data::CallbackDataType
+end
+
 """
     BasisStatusCode