  When instantons are put into the Higgs phase, vortices are attached to
instantons. We construct such composite solitons as 1/4 BPS states in
five-dimensional supersymmetric U(Nc) gauge theory with Nf(>=Nc) fundamental
hypermultiplets. We solve the hypermultiplet BPS equation and show that all 1/4
BPS solutions are generated by an Nc x Nf matrix which is holomorphic in two
complex variables, assuming the vector multiplet BPS equation does not give
additional moduli. We determine the total moduli space formed by topological
sectors patched together and work out the multi-instanton solution inside a
single vortex with complete moduli. Small instanton singularities are
interpreted as small sigma-model lump singularities inside the vortex. The
relation between monopoles and instantons in the Higgs phase is also clarified
as limits of calorons in the Higgs phase. Another type of instantons stuck at
an intersection of two vortices and dyonic instantons in the Higgs phase are
also discussed.
